text:
## Results (4 matches)

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 3 5 13
7800.2.a.p $$1$$ $$62.283$$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$-2$$ $$-$$ $$-$$ $$+$$ $$-$$ $$q+q^{3}-2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots$$
7800.2.a.w $$1$$ $$62.283$$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$4$$ $$-$$ $$-$$ $$+$$ $$-$$ $$q+q^{3}+4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots$$
7800.2.a.bb $$2$$ $$62.283$$ $$\Q(\sqrt{33})$$ None $$0$$ $$2$$ $$0$$ $$-1$$ $$-$$ $$-$$ $$+$$ $$-$$ $$q+q^{3}-\beta q^{7}+q^{9}-\beta q^{11}+q^{13}+\cdots$$
7800.2.a.bq $$3$$ $$62.283$$ 3.3.2700.1 None $$0$$ $$3$$ $$0$$ $$0$$ $$-$$ $$-$$ $$+$$ $$-$$ $$q+q^{3}-\beta _{1}q^{7}+q^{9}-\beta _{1}q^{11}+q^{13}+\cdots$$


text:
+0

# Equation

0
42
2

What is the sum of all possible solutions of $\frac{9x}{13}+\frac{13}{9x}=\frac{250}{117x} + \frac{3x}{2}$?

Aug 23, 2021

#1
0

Don't think i have the time to show the steps, but the sum of the possible solutions of that equation is 0.

Aug 23, 2021
#2
+45
0

You can move 3pi/2 and 9pi/13 to one side, and add them together, and then you can cross multiply, it's basic algebra.

Aug 23, 2021

format: plain


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402 views

Read the following information to answer the given questions.

Six lectures $A, B, C, D, E, F$  are to be organised only on one each day from Monday to Sunday in accordance with the following..

1. $C$ should not be organised on Friday.
2. $A$ should be organised immediately after $D$.
3. There should be a gap of two days between $B$ and $F$.
4. On one day there will be no lecture (Saturday is not that day), just before that day $F$ will be organised.
5. $E$ should be organised on Wednesday and should not be followed by $F$.

On which day will the lecture $D$ will be organised ?

1. Friday
2. Saturday
3. Sunday
4. Thursday

ans B

monday

tuesday     C

wednesday   E

thursday      F

friday      leave

saturday    D

sunday   A

by
182 points

1 vote
1
1 vote


text:
702 views

Let SHAM$_3$ be the problem of finding a Hamiltonian cycle in a graph $G=(V,E)$  with $|V|$  divisible by $3$ and DHAM$_3$  be the problem of determining if a Hamiltonian cycle exists in such graphs. Which one of the following is true?

1. Both DHAM$_3$ and SHAM$_3$ are NP-hard
2. SHAM$_3$ is NP-hard, but DHAM$_3$ is not
3. DHAM$_3$ is NP-hard, but SHAM$_3$ is not
4. Neither DHAM$_3$ nor SHAM$_3$ is NP-hard
recategorized | 702 views
0

Dirac's theorem and Ore's theorem are necessary but not sufficient. So 'determining if a Hamiltonian cycle exists' is also NPH.

The only difference between SHAM and DHAM, in SHAM |V| is divisible by $3$.. which can be check in constant amount of time.. So the hardness of the two problem will the same... Next, finding hamiltonian cycle comes under NPC problem and NPC problem is a subset of NPH, so both are NPH..

So, option (A)

edited by
0
|V| is divisible by 3 in both the problems. The difference between DHAM and SHAM is that one is about finding if the Hamiltonian cycle exist and the other one is about finding that cycle.
0
$SHAM_3$: "finding a Hamil..." does not sound as a decision problem. Isnt it? Yes finding if Hamiltonian circuit exists or not is NP-Complete and hence NP-Hard...Am I write?
0

SHAMcan be reduced to DHAM3 which is a decision problem. So it must be at least as hard as DHAM3.

0
really ? changing non decision problem into decision problem ?
0
@Mahesha999 Your are correct, SHAM is not really a decision problem and is not eligible for NP-hard or NP-complete.


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What is the answer for number 15 16 17

Question:

what is the answer for number 15 16 17

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Whitney gets an A- on her science test on ecology. She studied for several afternoons with a group of classmates and she says that the test was difficult but fair. Whitney's explanation for her good test grade is an example of A. an external attribution. B. an internal attribution. C. a cognitive belief. D. a behavioral attribution

Whitney gets an A- on her science test on ecology. She studied for several afternoons with a group of classmates and she says that the test was difficult but fair. Whitney's explanation for her good test grade is an example of A. an external attribution. B. an internal attribution. C. a cognitive be...

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You run a construction firm. You have just won a contract to build a government office building. Building it requires an investment of $10.4 million today, and$5.0 million in one year. The government will pay you \$24.0 million in one year upon the building's completion. Suppose the cash flows and t...

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The following question is based on your reading of "Dr Jekyll and Mr. Hyde." Which of the following best describes the appearance of Jekyll's home? a The beautiful exterior sharply contrasts with C The front of the house is elegant, but the back the dark, unwelcoming interior is in disrepair b. The windows are always open, but the shades d Laboratory equipment is visible from the are always drawn windows only at the front of the house

The following question is based on your reading of "Dr Jekyll and Mr. Hyde." Which of the following best describes the appearance of Jekyll's home? a The beautiful exterior sharply contrasts with C The front of the house is elegant, but the back the dark, unwelcoming interior is in disrepair b. The ...

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# Is there a closest free-return trajectory to a black hole?

The scenario I am imagining is a test particle falling from infinity with an initial off-center velocity. In Newtonian mechanics the path would be a hyperbola. But I assume that, near a black hole, that is no longer a good approximation.

I have read about the innermost stable circular orbit, and it seems like there might be an innermost free-return distance, where any object that falls within that distance will need to use rockets to get back out. I may have even come across this idea, but I am not finding it with a Google search.

I have an argument that there must be such a distance. Imagine that you could fly past a supermassive black hole just half a meter above the event horizon and continue on to deep space without using rockets. Tidal forces aren't so great at the event horizon for a SMBH, so there should be no reason you can't extend your arm just over half a meter down and not pull back a bloody stump. But you can't retrieve things from below the event horizon. This paradox is resolved if, in fact, the only way to escape from that close to the EH is to exert massive force using rockets (or some other method). Even though you are not yet inside the EH, your free fall path will take all of you there, and the force to prevent that would pulp you anyway. I think it is related enough to the main question to ask: Is this analysis correct?

• For a non-rotating BH, the ISCO radius is $3r_s$ and the photon sphere radius is $1.5r_s$, so even a photon with a hyperbolic trajectory can't approach the BH any closer than that. That article also mentions a marginally bound orbit at $2r_s$, but it doesn't give any details. I guess it may be the limit of a hyperbolic / parabolic trajectory for massive particles. Things are more complicated for spinning black holes, and SMBHs tend to have a lot of spin. See astronomy.stackexchange.com/a/20292/16685 – PM 2Ring Apr 11 at 21:03

The trajectory of a ballistic body, whether in Newtonian or Relativistic physics depends on the initial energy and angular momentum.

The difference is that in Newtonian physics, if the mass is compact enough, the infalling object (if given an initial kinetic energy) will never hit the central object, unless it is a direct hit, and will scatter off to infinity.

This can also happen in GR, but only if the specific angular momentum exceeds a threshold that increases with the initial energy of the particle, but can never be less $$4GM/c$$, and only as small as this for an object that falls from infinity with no initial kinetic energy. If the specific angular momentum is less than this then the infalling object will always end up in the black hole (for a freefalling trajectory).

For the limiting case, the closest approach to the black hole is $$2r_s$$, which is the local maximum of the effective potential.

However, the body can get closer if it has more angular momentum and more initial energy. If we allow the angular momentum and energy to get arbitrarily large then the body can get to $$r=1.5 r_s$$.

So I think this is the answer you are looking for. This is the value for a Schwarzschild black hole. For a spinning black hole, things are more complicated.

EDIT: Some details

The effective potential in the Schwarzschild metric is given by $$\frac{V_{\rm eff}}{mc^2} = -\frac{r_s}{2r} + \frac{L^2}{2m^2 r^2 c^2} - \frac{r_sL^2}{2m^2r^3c^2}\ ,$$ where $$r_s$$ is the Schwarzschild radius and $$L/m$$ is the specific angular momentum of the free body.

The radial equation of motion in terms of the proper time $$\tau$$ of the body is $$\frac{1}{2c^2}\left( \frac{dr}{d\tau}\right)^2 = \frac{1}{2}\left( \frac{E^2}{m^2 c^4} -1 \right) - \frac{V_{\rm eff}}{mc^2} \ .$$

To find the local maximum of the effective potential, differentiate with respect to $$r$$ and equate to zero. This gives you that the specific angular momentum at the maximum is $$\frac{L^2}{m^2} = \frac{c^2 r_s r^2}{2r - 3r_s}\ .$$

Allowing the energy of the body to be $$E = \gamma mc^2$$ (where $$\gamma$$ is the Lorentz factor) one can set $$dr/d\tau = 0$$ at the scattering minimum radius and then subsititute in the value for $$L^2/m^2$$ at the maximum of the effective potential. The rationale here is that the body will get to its closest point when it has an energy corresponding to the maximum in the effective potential. If it has any more energy than this it will just plunge into the black hole. Any less and it will scatter at a larger radius.

This condition gives you a quadratic equation in $$r$$ which can be solved in terms of $$\gamma$$. If $$\gamma=1$$ then the quadratic becomes a linear equation with $$r=2r_s$$. This corresponds to the case of a particle falling in from rest at infinity and meeting a local maximum of the potential of $$V_{\rm eff}=0$$. The body then scatters off to (just) infinity (i.e. the marginally bound case). This is illustrated below, where the horizontal purple line is a line corresponding to an energy $$E/mc^2=1$$. The peak at B is at $$r=2r_s$$ where $$V_{\rm eff}=0$$ (I've also shown the equivalent Newtonian potential for contrast).

If you allow $$\gamma$$ to be larger then the the viable solutions for the minimum $$r$$ lie in the bounds $$1.5 r_s \leq r \leq 2r_s$$. If $$\gamma \rightarrow \infty$$ then the scattering minimum radius $$r \rightarrow 1.5r_s$$, which also requires $$L/m \rightarrow \infty$$. The plot below shows the potential for $$L/m = 10GM/c$$, with a purple line representing a body with $$E/mc^2=2$$ which would scatter to infinity. The peak at B is just above $$r=1.5r_s$$.

If you want to play with this (my) Geogebra widget, you can find it at https://www.geogebra.org/m/u7y6huer

• The closest you can approach a Schwarzschild BH on a freefalling trajectory and still return to infinity is $\tfrac{3}{2}r_s$. The $2r_s$ you mentioned is closest approach for a marginally bound trajectory. – mmeent Apr 11 at 21:43
• @mmeent Correct. I have got myself in a tangle with the limiting case of a minimum angular momentum. – ProfRob Apr 11 at 22:24
• If there is a link to some math on this subject it would perhaps be useful. – StephenG Apr 12 at 14:36
• I was about to ask some clarifying questions but then I reread the answer and I think you've made edits that addressed my questions already. Thanks! – Mark Foskey Apr 12 at 23:28
• Thanks for the great edit and also the link to geogebra.com which I was not aware of until now. – StephenG Apr 13 at 12:11


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# 2nd Year Math Chapter 7 Notes | Unit 07: Vectors

 2nd Year Math Chapter 7 Notes
2nd Year Math Chapter 7 Notes | Unit 07: Vectors are being uploaded here. These F.Sc. Math part 2 chapter no. 7: Vectors notes are in accordance with the syllabus and paper pattern of Punjab Text Book Board Lahore. These notes are very useful for the students belong to the Sargodha board, Lahore Board, Sahiwal Board, Multan Board, Kashmir board, Rawalpindi board, Gujranwala board, and DG Khan board.

### 2nd Year Math Chapter 7 Notes | Unit 07: Vectors

You can easily view or download these notes. If you want to view 12th class Math chapter 7 notes online, there must be a PDF reader installed on your computer. These notes cover the following topics
• Introduction
• Geometric Interpretation of Vector
• Multiplication of Vector by a Scalar
• Addition and Subtraction of two vectors
• Position Vector
• Vector in Plane
• Properties of Magnitude of a Vector
• Another Notation for representing Vectors in Plane
• AUnit Vector in the Direction of another given Vector
• The Ratio Formula
• Vector Geometry
• Introduction of Vector in Space
• Concept of a Vector in Space
• Properties of Vector
• Another notation for representing Vectors in Space
• Distance between two Points in Space
• Direction Angles and Direction of a Vector
• The Scalar Product of Two Vectors
• Deductions of the Important Results
• Perpendicular (Orthogonal) Vectors
• Properties of Dot Product
• Analytically Expression of Dot Product $u\cdot v$
• Angle between two Vectors
• Projection of One Vector upon another Vector
• The Cross Product or Vector Product of two Vectors
• Derivation of useful results of Cross Product
• Properties of Cross Product
• Analytically Expression of $u×v$
• Parallel Vector
• Area of Parallelogram
• Area of Triangle
• Scalar Triple Product of Vectors
• Analytically Expression of $u\cdot \left(v×w\right)$
• The Volume of the Tetrahedron
• Application of Vectors in Physics and Engineering
To view online or download 2nd year Math chapter 7 notes in PDF Format, click the button given below. Don't forget to share these notes with your friend, class fellows, and teachers.


text:
Test whether elements in vector x are effectively zero, within the square root of machine tolerance for a floating point number.

almostZero(x, y = 0, tolerance = sqrt(.Machine\$double.eps))

## Arguments

x Numeric vector of values to compared against 0 Value to be compared against. Default is 0. The maximum difference between x and y necessary to call a value non-zero

## Value

TRUE or FALSE for each element of the vector x.

## Details

Returns TRUE for all vector elements that are within rounding error of zero.


text:
Abstract

The COVID-19 pandemic has led to large disruptions to the Australian labour
market. Initially, workers were less likely to change jobs because of the uncertain economic environment, the
decrease in advertised jobs and the JobKeeper program that helped workers remain attached to their employers.
More recently, job mobility has increased as workers have caught up on planned job changes or been encouraged
by the strong labour market to change jobs, particularly in high-skilled roles experiencing strong labour
demand. This article reviews developments in job mobility in Australia through the pandemic, and compares
these outcomes to other advanced economies. It also examines the potential implications for wages; a high
rate of job mobility tends to be associated with higher wages growth in a tight labour market, as employers
in sectors with high demand for labour compete for new staff or raise wages to retain staff.

Introduction

Job mobility – the movement of workers between jobs – underpins the efficient operation of the
labour market by matching people with jobs that better fit their preferences and skills. People switch
jobs for a number of reasons. These can include higher wages, improved working conditions, different
hours, workplace flexibility, job satisfaction or job security. To the extent that job mobility results
in better job matching and increased labour reallocation to more productive firms, it can play a role in
labour productivity gains. Job mobility is also a key feature of labour market flexibility that helps the
economy adjust to economic shocks and structural change, which can affect the number and types of jobs
available.

Prior to the pandemic, the Australian labour market had experienced a downward trend in job mobility for a
number of decades, and job mobility was around historically low levels (Graph 1). While low job
mobility is not necessarily problematic – some job changes are not voluntary and there can be
benefits from longer job tenure – the decline in job mobility may have been associated with a
general decline in business dynamism (Ellis 2021). In particular, it coincided with a slower reallocation
of labour to productive firms, which contributed to lower productivity growth (Andrews and Hansell 2019).
The rate of job mobility tends to move with the business cycle, with sharp declines experienced in the
recessions of the early 1980s and 1990s and following the global financial crisis (GFC). The step down in
mobility following the latter two downturns suggests that structural factors are also usually at play.

The onset of the COVID-19 pandemic led to large disruptions to the Australian
labour market that corresponded with an initial decline in economic activity amid restrictions to contain
the health effects of the virus. A significant number of people were stood down from their jobs or faced
reduced hours during this period. There was also a sharp fall in job mobility, reflecting a reluctance to
change jobs, fewer opportunities to switch jobs and the effect of JobKeeper in keeping workers linked to
their employers. As economic activity rebounded in late 2020, so too did the labour market. As of early
2022, job mobility appears to be around its highest level in over a decade, returning to levels last seen
prior to the GFC. The increase has been underpinned by the strength of the labour market, especially in
particular sectors, and a backlog of job changes that had been deferred.

This article reviews developments in job mobility and turnover in Australia since the onset of the
pandemic, and compares these outcomes to other advanced economies. It also examines the role of job
mobility as a potential factor influencing wage outcomes, which can have implications for monetary
policy.

Measuring job mobility and turnover

Aggregate measures of job mobility and job turnover in Australia are published by the Australian Bureau of
Statistics (ABS) on an annual basis as a supplement to the monthly Labour Force Survey (LFS) and are
available from 1972. To examine job mobility and turnover on a timely basis through the pandemic, we made
use of several quarterly measures in the person-level Longitudinal Labour Force Survey (LLFS) microdata,
which covers LFS responses from 1982. The dataset followed (anonymised) individuals for
eight consecutive months, which allowed us to track job switching behaviour over that time. The key
metrics of interest are:

• Job turnover: people that lost or left any job in the past three months. Job
turnover can be voluntary or involuntary. Voluntary job separations capture workers who leave a job
with the intention of finding another job, as well as those who quit for life-cycle or other personal
reasons. Involuntary separations include workers who lose a job due to economic reasons (such as
retrenchments), dismissals, the ending of a temporary job and own illness.
• Job mobility rate (constructed by the Reserve Bank): the share of employed persons
who have been working for their current employer/business for less than three months and were also
employed in the previous quarter. The job mobility rate is a narrower measure than
job turnover; job mobility captures workers who left their jobs and are currently employed in a new
one, including those who started a new job after a brief period of unemployment or absence from the
labour force. The higher frequency measure of job mobility tracks relatively closely with the annual
ABS data, despite conceptual differences between the measures; a person who switches jobs multiple
times during the year would only be counted once in the annual measure.
• Expected job mobility rate: the share of workers who expect to change jobs in the
next 12 months.

Job mobility at the onset of the pandemic

The onset of the COVID-19 pandemic in Australia in March 2020 had significant
effects on the labour market (Graph 2). Movement restrictions and lockdowns, as well as
precautionary behaviour, meant many businesses closed or operated at reduced capacity. Corresponding to
the decline in economic activity, a significant number of workers lost their jobs or had their hours
sharply. Government policy measures, including the introduction of the JobKeeper wage subsidy, helped to
keep many workers attached to their employer; in the absence of JobKeeper, employment would have declined
much further (Bishop and Day 2020).

These disruptions had a large impact on job turnover (Graph 3). The number of employed persons who
lost their jobs rose sharply. This included workers who were retrenched or made redundant, and those who
lost their jobs because their employer went out of business or had no work available. This was associated
with a large increase in the number of people temporarily exiting the labour force, with a
3.3 percentage point drop in the participation rate in the three months to May 2020.

Job mobility also declined at the start of the pandemic. The share of employed persons who changed jobs
fell to its lowest level in over two decades in mid-2020; only around 2 per cent of employed
people changed jobs in the three months to August 2020 (Graph 1). Seeking out a better job/wanting a
change became a much less common reason for leaving a job. This likely reflected workers’ increased
risk aversion to switching jobs amid heightened uncertainty, as well as fewer opportunities for workers
seeking to improve their job match due to lower labour demand. In broad terms, this is consistent with
the pro-cyclical nature of voluntary turnover. Voluntary turnover tends to fall during economic downturns
when workers are less confident about switching jobs and employment opportunities are less plentiful; it
tends to rise during periods of strong labour market conditions when workers are more willing to bear the
costs and risks of changing jobs. However, this wasn’t a typical cyclical event as the downturn was
precipitated by lockdown measures to counter the health effects of the pandemic. The extended lockdowns
also meant it was more difficult for workers to look for new jobs, and the implementation of JobKeeper
kept many workers linked to their employers. With that said, the number of people who left their jobs due
to ‘other reasons’ spiked during periods of lockdowns, which may have captured workers who
left their jobs during the pandemic for a multitude of reasons, including wanting a better job, but felt
that it was too difficult to attribute their situation to one category.

The decline in job switching was evident for both full-time and part-time workers, but more so for the
latter group. This likely reflected the more adverse impact of lockdowns on contact-intensive industries
that typically employ a higher proportion of part-time workers and casuals, such as accommodation &
food services (discussed below).

The recovery in job mobility

As health-related restrictions eased and the economic recovery progressed, the number of people working
reduced hours was gradually unwound and more workers gained employment as firms resumed hiring (or
workers resumed their previous jobs if they returned to their previous employer). Many workers who had
left the labour force at the height of the pandemic have since returned. Consistent with the improvement
in labour market conditions, the number of involuntary job separations has declined to be around more
usual levels. Voluntary resignations have picked up, with the most common reason being to seek a better
job or the desire for a change. This is also evident in the increase in the job mobility rate, which
since mid-2021 has been at a high level relative to recent history. For example, the average job mobility
rate over the past decade was 2.8 per cent; in February 2022, it was 3.3 per cent
– nearly 440,000 people switched jobs in the three months to February. In contrast to the
earlier stages in the pandemic when the mobility rate declined more for part-time workers, it is now at a
similarly high level for both full-time and part-time workers.

Information from the Bank’s liaison program also indicates that voluntary turnover rates have
increased since mid-2021 to be above average, particularly in sectors or roles experiencing strong demand
for labour. According to liaison, while this partly reflects some catch-up following two years of lower
staff turnover during the pandemic, it is also due to increased competition for labour, with workers
being enticed to join other firms for a higher salary. A simple counterfactual exercise – which
compares the actual cumulative flow of job switches since February 2020 to what the cumulative flow would
have been if the number of job switches had remained at its average level in 2019 throughout the pandemic
– suggests there has been a sizeable number of job switches beyond a simple catch-up following the
initial decline, accounting for around 1.5 per cent of total employment as of February 2022.

The speed of recovery in job mobility from the COVID-19 shock is in contrast
to the GFC – job mobility did not recover to its pre-GFC levels and trended lower for the next
decade. The natures of these events, however, were distinctly different. The recent downturn was
precipitated by lockdown measures to counter the health effects of the virus. The rapid recovery in the
labour market that followed the removal of these measures and ongoing underlying momentum supported by
policy settings have enabled a faster rebound in job mobility, which quickly exceeded its pre-pandemic
level. The labour market recovery was much slower after the GFC and aggregate demand remained low for
many years. A similar post-GFC decline in job mobility has been evident in other advanced economies,
suggesting that structural factors play a role – for example, this may be related to the ageing
population, the decline in the share of startups and a rise in larger firms, and policy settings (Engbom
2019; Decker et al 2014; Decker et al 2020; Hermansen 2019).

Characteristics of job mobility during the pandemic

The impacts of the pandemic on the labour market have been uneven across different groups of people and
sectors in the economy. Activity restrictions and isolation requirements have more adversely affected
industries that tend to be contact intensive, such as accommodation & food services. These industries
also tend to employ a higher share of younger, part-time and casual workers. By contrast, employment has
recovered rapidly to be well above pre-pandemic levels in other industries, such as health care &
social assistance and professional services; this is partly due to the health response to the pandemic,
strong underlying demand and/or the availability of remote or socially distanced work.

While the extent of job mobility generally varies across industries, the impact of the pandemic on job
mobility has also been more concentrated in some areas of the economy than others. In general, industries
with lower earnings, lower wages growth and a younger average worker age are typically associated with
higher job mobility. This is consistent with workers having less incentive to move from their existing
job or industry in which they have accumulated experience that adds to their earning potential
(D’Arcy et al 2012). High-skilled jobs have experienced particularly sharp increases in
job mobility since mid-2021; notably, the job mobility rate in professional services has increased to be
around its highest level in over two decades (Graph 4). This is consistent with information from the
Bank’s liaison program that suggests that strong labour demand in professional services has
encouraged more employees to seek higher rates of pay or new opportunities, and resulted in a very
competitive job market for businesses seeking to hire workers, particularly those skilled in IT. Job
mobility in the healthcare industry – where employment growth has been particularly strong since
the start of the pandemic – has also increased to be at its highest level in over two decades.
Likewise, job mobility in contact-intensive industries and construction has recently picked up, after
being relatively low during the earlier stages of the pandemic.

Changes in job mobility during the pandemic have been evident across most age groups (Graph 5). At
the onset of the pandemic, the decline in job mobility was most pronounced among younger workers (aged
15–19 years and 20–29 years), likely reflecting reduced demand for entry-level
workers and the more adverse impact on contact-intensive industries that employ a higher share of young
workers. Typically, younger workers have higher rates of job switching than older workers. This is
because young people have less firm- and industry-specific human capital than more experienced older
workers and so have more to gain by changing jobs and increasing the quality of a job match; an example
of this is a university graduate who switches from casual employment to a full-time career in an industry
related to their studies. Job mobility for these workers has since recovered, alongside improvements in
youth labour market outcomes. In fact, job mobility has increased for most age groups in recent months to
be above pre-pandemic levels.

Job mobility includes workers changing jobs within the same sector (referred to as ‘churn’)
and workers switching to jobs in different sectors (Graph 6). In the early stages of the pandemic,
there was a sharper decline in job churn than in workers changing industries, though the latter also
declined. Since mid-2021, as job mobility increased with the strengthening labour market, the proportion
of job switches involving changing industries as distinct from churn has returned to its historical
average. However, there have been some differences across sectors. In the business services sector, the
large increase in job mobility from mid-2021 was mostly driven by increases in job churn, with only a
modest increase in workers switching jobs from outside the sector. The household services sector has
experienced similar increases in job switches within and into this sector, while job mobility in the
goods sector has been relatively stable throughout the pandemic. In addition, data on job switching by
the skill level of the job shows that those workers who have changed jobs have mostly moved into a job
requiring the same level of skill.

Job switching expectations

Some insights on whether job mobility will remain elevated or return to previous levels can be obtained
from data on future employment expectations. Actual job mobility has largely followed expected job
mobility over the past two decades, although these have differed at times (Graph 7). Workers remain
upbeat about the jobs outlook, with the share of workers expecting to change jobs/seek other employment
within the next year being around its highest level since 2008. The increase in expected job mobility has
been primarily driven by full-time workers and is more pronounced among female full-time workers. By
contrast, the share of part-time workers expecting to change jobs has returned to its longer term
downwards trajectory.

Higher skilled occupations have continued to report elevated levels of expected job mobility, while lower
skilled occupations have remained more in line with the longer term average. By industry, expected job
mobility for the year ahead was elevated in healthcare & social assistance and financial &
insurance services, and subdued in accommodation & food services, arts & recreation and other
services, and construction.

Job mobility and wages growth

While workers may change jobs for a number of reasons, higher wages are often the motivation. Data from
the Household, Income and Labour Dynamics in Australia (HILDA) survey show that people who move jobs
generally gain higher-than-average wage increases (Graph 8). People who stay with the same employer
have lower but more stable wage growth over time. Recent domestic and international research has shown
that higher job mobility also tends to be associated with higher aggregate wages growth
(Faberman and Justiniano 2015; Karahan et al 2017; Moscarini and Postel-Vinay 2017;
Deutscher 2019). This can arise from two channels: directly, because workers typically experience a pay
bump from changing jobs; and indirectly, because an employer may offer a pay raise to retain a worker in
their current job due to competition for labour.

In Australia, higher job mobility has tended to be associated with higher aggregate wages as measured by
the Wage Price index (WPI) (notwithstanding that other factors are also important drivers of wages, as
discussed below) (Graph 9). The WPI measures the wages of a sample of jobs (rather than a sample of
workers). In this way, it will capture the higher wages that an employer pays for a particular job in the
face of high job mobility – for example, higher wages offered to poach or retain a worker. While
high job mobility tends to be associated with wages growth, the direction of causality could run either
or both ways. High job mobility may lead to workers receiving a wage increase by switching jobs or
through higher competition for workers in their industry. However, the reverse could also occur, whereby
wages growth incentivises workers to switch jobs within their industry or it could lead to higher job
mobility across industries – for example, higher wages growth in particular sectors may lead
workers to retrain and switch jobs into those sectors, and/or encourage firms to consider workers coming
from other industries even if their work experience is not seen as directly relevant.

As part of its set of models, the Reserve Bank uses the Phillips curve framework to consider the
implications for wages growth based on a number of cyclical factors, including: spare capacity in the
labour market; inflation expectations; a measure of changes in firms’ output prices; and lagged wages
growth (Bishop and Greenland 2021). Job mobility is not usually included in this model; however, using
this framework, we explored whether job mobility provides additional information for future wages growth
above and beyond the variables in the Bank’s baseline wages model (see Appendix A for full model
results).

The results indicate that job mobility can help to explain future changes in WPI growth in the near term.
According to the model, the increase in the job mobility rate in the decade leading up to the GFC
supported wages growth generally, and the trend decline in job mobility over the following decade is
estimated to have been a drag on wages growth. With that said, the inclusion of job mobility in the model
provides some but not much additional information to explain wages growth beyond the baseline model. The
existing measure of spare capacity in the labour market in the model (the unemployment gap) is able to
account for most of the variation in wages growth related to labour market tightness, including the
effects of job mobility. This result is not too surprising as the unemployment gap and job mobility tend
to move closely together.

Overall, the recent pick-up in job mobility in some sectors in Australia is expected to contribute over
time to employers offering higher wages to retain workers or hire new workers as the labour market
tightens. Some firms in the Bank’s liaison program reported having increased wages in response to
elevated turnover, with some of these firms paying out-of-cycle wage increases to some workers to meet
the increase in market salaries for those roles and prevent further increases in turnover. In addition,
some firms have paid more to attract new staff to fill vacant or new roles.

Job turnover, mobility and wages growth through the pandemic: Comparing Australia with the United States and the United Kingdom

Similar to Australia, job turnover and mobility have increased sharply in recent months in the United
States and the United Kingdom (Graph 10). In the United States, where data on job mobility are not
readily available, the job quits rate – the share of workers leaving their jobs to take new jobs or
to exit the labour force – has been around a record high since mid-2021. In the United Kingdom, the
job mobility rate has increased to an elevated level. Wages growth has picked up sharply in both of these
countries – notably more so than in Australia. Notwithstanding the structural differences in the US
and UK labour markets, the extent of the recovery in labour supply, in the face of strong labour demand,
appears to have played a role in the different wage growth outcomes so far. While labour force
participation declined at the onset of the pandemic in all three countries, it has since increased in
Australia to be at a record high, whereas the US and UK participation rates still remain below
pre-pandemic levels (although it has been recovering recently in the United States) (Graph 11).
COVID-19-related health concerns, high accumulated savings due to fiscal
support, increased retirements and workers re-evaluating longer term personal and professional goals
(perhaps as part of the ‘Great Resignation’) have likely contributed to people leaving the
labour force in the United States and the United Kingdom (Agarwal and Bishop 2022). As a result, firms in
these countries have been paying higher wages to retain and attract workers. Conversely, health concerns
appear to have had less of an impact on labour supply in Australia – partly due to comparatively
better health outcomes – and retirements have remained around usual levels throughout the pandemic.

Conclusion

Job mobility in Australia has increased to a high level recently, underpinned by the strong labour market.
This followed a sharp decline in job switching at the onset of the pandemic amid widespread disruptions
to the labour market, and a multi-decade trend decline prior. The increase in job switching appears to be
partly driven by strong labour demand for some high-skilled jobs, coupled with workers catching up on
planned job changes that were put on hold during the pandemic. High levels of job switching tend to be
associated with higher wages, both at the individual and aggregate level, in a tight labour market. While
job turnover has also been high in the United States and the United Kingdom, labour force participation
rates have not yet recovered to pre-pandemic levels; by contrast, Australia is experiencing a record high
level of participation. This appears to have played a role in relative wage outcomes across the
countries. The increased rates of job mobility in Australia in some sectors is expected to contribute to
higher labour costs as firms find that they need to offer higher wages and other benefits to attract new
workers or retain existing staff.

Appendix A: The RBA Wages Phillips curve model with job mobility

We examined whether the addition of the job mobility rate to the RBA Wages Phillips curve model provides
additional information for wages growth. We estimated the equation below:

$%∆{WPI}_{t}=\alpha +\beta %∆{WPI}_{t–1}+\gamma \left(\frac{{u}_{t–1}–{u}_{t–1}^{*}}{{u}_{t–1}}\right)+\tau ∆{u}_{t–1}+\phi \frac{{%{∆}^{ye}DFDdef}_{t}}{4}+\theta \frac{{Inflationexp}_{t–1}}{4}+\delta Jobmobilit{y}_{t–1}+{ϵ}_{t}$

Where:

 ${\mathrm{%}∆WPI}_{t}$ Quarterly growth in the private sector Wage Price Index (WPI) (hourly rates excluding bonuses); seasonally adjusted. ${u}_{t}$ Quarter-average unemployment rate; seasonally adjusted. ${u}_{t}^{\mathrm{*}}$ Non-accelerating inflation rate of unemployment (NAIRU) (described in Cusbert 2017); two-sided smoothed quarterly estimate. $∆{u}_{t}$ Quarterly change in the unemployment rate; seasonally adjusted. ${\mathrm{%}∆}^{ye}{DFDdef}_{t}$ Year-ended growth in the domestic final demand implicit price deflator; seasonally adjusted. ${Inflationexp}_{t}$ Trend inflation expectations estimated using a Kalman filter (described in Cusbert 2017); two-sided smoothed quarterly estimate. $Jobmobilit{y}_{t}$ Quarterly job mobility rate; seasonally adjusted.

Baseline model Job mobility 0.066***(0.020) −0.362**(0.169) 0.425***(0.109) 0.337***(0.110) −0.533***(0.156) −0.402**(0.193) −0.097(0.073) −0.059(0.073) −0.012(0.086) −0.109(0.102) 0.587***(0.132) 0.771***(0.153) – 0.142**(0.057) 0.74 0.77 (a) The dependent variable is the quarterly growth in the private sector WPI (hourly rates excluding bonuses); standard errors in parentheses; ***, **, and * denote statistical significance at the 1, 5, and 10 per cent levels, respectively. Sources: ABS; authors’ calculations

References

Agarwal N and J Bishop (2022), ‘COVID-19 Health Risks and Labour Supply’, RBA
Bulletin, March.

Andrews D and D Hansell (2019), ‘Productivity‐Enhancing Labour Reallocation in
Australia’, Treasury Working Paper No 2019-06.

Bishop J and I Day (2020), ‘How Many
Jobs Did JobKeeper Keep?
’, RBA Research Discussion Paper No 2020-07.

Bishop J and E Greenland (2021), ‘Is
the Phillips Curve Still a Curve? Evidence from the Regions
’, RBA Research Discussion Paper
No 2021-09.

Cusbert T (2017), ‘Estimating the
NAIRU and the Unemployment Gap
’, RBA Bulletin, June, pp 13–22.

D’Arcy P, L Gustafsson, C Lewis and T Wiltshire (2012), ‘Labour Market Turnover and Mobility’, RBA
Bulletin, December, pp 1–12.

Decker R, J Haltiwanger, R Jarmin and J Miranda (2014), ‘The Role of
Entrepreneurship in US Job Creation and Economic Dynamism’, Journal of Economic
Perspectives
, 28(3), pp 3–24.

Decker RA, J Haltiwanger, RS Jarmin and J Miranda (2020), ‘Changing Business
Dynamism and Productivity: Shocks versus Responsiveness’, American Economic Review,
110(12), pp 3952–3990.

Deutscher N (2019), ‘Job-to-Job Transitions and the Wages of Australian
Workers’, Treasury Working Paper No 2019-07.

Ellis L (2021), ‘Innovation and
Dynamism in the Post-pandemic World
’, Speech to the Committee for the Economic Development
of Australia, Webinar, 18 November.

Engbom N (2019), ‘Firm and Worker Dynamics in an Aging Labor Market’, Federal
Reserve Bank of Minneapolis Working Paper No 756.

Faberman RJ and A Justiniano (2015), ‘Job Switching and Wage Growth’, Chicago
Fed Letter No 337.

Hermansen M (2019), ‘Occupational Licensing and Job Mobility in the United
States’, OECD Economics Department Working Paper No 1585.

Karahan F, R Michaels, B Pugsley, A Şahin and R Schuh (2017), ‘Do Job-to-Job
Transitions Drive Wage Fluctuations over the Business Cycle?’, American Economic
Review
, 107(5), pp 353–357.

Moscarini G and F Postel-Vinay (2017), ‘The Relative Power of
Employment-to-Employment Reallocation and Unemployment Exits in Predicting Wage Growth’,
American Economic Review, 107(5), pp 364–368.

HILDA Disclaimer


text:
## LSU Doctoral Dissertations

#### Degree

Doctor of Philosophy (PhD)

#### Department

Geography and Anthropology

Dissertation

#### Abstract

As sea levels rise, coastal trees are increasingly stressed by saltwater inundation, and stressed trees are ideal for studying changes in the environment through time. Dendrochronology has been used for more than a century to explain changes in past environments using tree rings. Using ring widths and stable isotopes from coastal trees, the effects of tropical cyclones and freshwater moisture availability are shown in regard to tree growth and ultimately are discussed in terms of change to the environment as a whole. Changes in tree growth as they relate to various climate parameters are measured using superposed epoch analysis, moving-window correlations, and spatial correlations.

Results show that tropical cyclones cause rapid declines in tree growth. In the first two years following large storm surges ($>$2.0 m), trees reduce growth in the stem as much as 15\%, whereas inland urban trees only show major growth declines in years following tropical cyclones with hurricane-force wind speeds ($>$33 ms$^{-1}$). For coastal trees that do not show strong tropical cyclone signals, drought conditions play a key role in growth declines. For example, tree-ring carbon isotopes show strongest spatial correlations (\textit{r}$>$0.5, \textit{p}\$

Coastal trees inundated by saltwater experience declines in growth following pulses such as drought, tropical cyclones, and anthropogenic disturbances. Future land management of coastal forests should consider changes in freshwater soil moisture availability when projecting changes in the environment. In summary, with accelerating sea-level rise and increasing tropical cyclone intensity, coastal areas can expect to see declines in tree growth.

Trepanier, Jill


text:
$$\renewcommand\AA{\unicode{x212B}}$$

# SortXAxis v1¶

## Summary¶

Clones the input MatrixWorkspace(s) and orders the x-axis in an ascending or descending fashion.

## Properties¶

Name

Direction

Type

Default

Description

InputWorkspace

Input

MatrixWorkspace

Mandatory

Input Workspace

OutputWorkspace

Output

MatrixWorkspace

Mandatory

Sorted Output Workspace

Ordering

Input

string

Ascending

Ascending or descending sorting. Allowed values: [‘Ascending’, ‘Descending’]

## Description¶

Clones the input Matrix Workspaces and orders the x-axis in an ascending or descending fashion. Ensures that the y-axis and error data as well as optional Dx data are sorted in a consistent way with the x-axis.

This algorithm is for use with either point data based workspaces or histogram based workspaces. It is particularly suitable for reformatting workspaces loaded via LoadAscii.

## Usage¶

#Import requirements
import numpy as np

# Create the workspace
dataX = [2., 1., 3.]
dataY = [2., 1., 3.]
dataE = [2., 1., 3.]
ws = CreateWorkspace(DataX=dataX,DataY=dataY,DataE=dataE,UnitX='TOF',Distribution=True)

# Print out the "Unordered X Axis"
print("Unordered Print")

# Sort the X Axis in a Descending fashion
ws = SortXAxis(InputWorkspace='ws', Ordering='Descending')
print("In order print: Descending")

# Sort the X Axis in a Ascending fashion
ws = SortXAxis(InputWorkspace='ws', Ordering='Ascending')
print("In order print: Ascending")

Output:

Unordered Print
[ 2.  1.  3.]
[ 2.  1.  3.]
[ 2.  1.  3.]
In order print: Descending
[ 3.  2.  1.]
[ 3.  2.  1.]
[ 3.  2.  1.]
In order print: Ascending
[ 1.  2.  3.]
[ 1.  2.  3.]
[ 1.  2.  3.]

Categories: AlgorithmIndex | Transforms\Axes | Utility\Sorting


text:
# Runic crescent wrench

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Runic crescent wrench
Weapon information
Precise dimensions 2 x 3/12
Material klatchian steel
Weight 3 3/9
Thaums/sec 6 stable / 7 talisman / 12 max
Hands 1
Commands Smash Crush
Melee type Mace
Judge data
Speed 9 Very fast
Maximum damage 9 Pretty high
Average damage 6 Quite low
 Str Overall Attack ease Parry ease
 13 14 excellent 9 pretty easy 9 pretty easy 14 14 excellent 10 easy 10 easy 15 14 excellent 10 easy 10 easy 16 14 excellent 10 easy 11 rather easy 17 14 excellent 10 easy 11 rather easy 18 14 excellent 10 easy 11 rather easy 19 14 excellent 10 easy 12 very easy 20 14 excellent 10 easy 12 very easy 21 14 excellent 11 rather easy 12 very easy 22 14 excellent 11 rather easy 13 extremely easy 23 14 excellent 11 rather easy 14 incredibly easy

## Long Description

Two feet long and forged from fine Gebra steel, this wrench is truly an Artificer's tool.  Along its length, runes inlaid in fine silver wire spell out a message for the initiated, and a constellation of silver stars around the crescent head twinkle in any available light, and perhaps in none at all.  The grip seems to shape itself to the wielder's hand, and the balance is light and wieldy.
There are eleven gleaming stars in the constellation.
It appears to have something written on it.

Written in silvery runes in Wizard_spells:
If thy heart be that of a true Artificer, thou mayst invoke this rod for aid with mechanisms arcane and technologick.

## Appraises As

The runic crescent wrench is about two feet long and three inches wide. It is made of klatchian steel and could be used as a weapon of type mace.

## Notes

Roughly a minimum 200 bonus in magic.items.held.rod is required to invoke this item with low success chance, but closer to 230+ bonus for better success rates is desired, in order to use it to complete the Achievement: Steampunk Artificer achievement.

Also, the act of invoking the wrench requires 300 magic points.


text:
# Multiple testing and logistic regression

I want to perform a number of univariate regressions with different symptoms (e.g. fever, cough, sneezing) as the response variable and one categorical variable (which is always the same each time) as an explanatory variable - age group (0-4, 5-14, 15-64, 65+). In total there are 18 symptoms so hence, I want to do 18 regressions and want to correct for multiple testing using Holm's method.

For each regression I get 4 p-values - one overall age group p-value and 3 p-values comparing individual age groups to a reference age group (15-64). I am unsure about how to calculate confidence intervals and adjusted p-values using Holm's method in this scenario. Does anybody know how to do this?

The R rms package supports general simultaneous confidence intervals. The analyst can specify a series of comparisons, and simultaneous adjustment will be done over all the current set of comparisons. For more information see the rms contrast.rms function, along with Predict and other functions. Here's an example using ols. Substitute lrm for ols to do binary or proportional odds logistic regression.

require(rms)
set.seed(1)
n <- 800
treat <- factor(sample(c('drug','placebo'), n,TRUE))
sex   <- factor(sample(c('female','male'),  n,TRUE))
age   <- rnorm(n, 50, 10)
y     <- .05*age + (sex=='female')*(treat=='drug')*.05*abs(age-50) + rnorm(n)
f     <- ols(y ~ rcs(age,4)*treat*sex)

# show separate estimates by treatment and sex

plot(Predict(f, age, treat, sex='female'))
plot(Predict(f, age, treat, sex='male'))
ages  <- seq(35,65,by=5); sexes <- c('female','male')
w     <- contrast(f, list(treat='drug',    age=ages, sex=sexes),
list(treat='placebo', age=ages, sex=sexes),
conf.type='simultaneous'))
xYplot(Cbind(Contrast, Lower, Upper) ~ age | sex, data=w,
ylab='Drug - Placebo')
xYplot(Cbind(Contrast, Lower, Upper) ~ age, groups=sex, data=w,
ylab='Drug - Placebo', method='alt bars')

For confidence interval, I think you can just use the logistic regression outputs. For adjust p-values, refer to this wiki link: http://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method

To my understanding, you are first test an overall age group, then testing a nested hypothesis. I would recommend you test overall group effects first, then go deeper to test individual groups.

• No, 95% confidence intervals are constructed such that, with a single data sample, the actual point is within the interval with probability $0.05$. It does not account for multiple testing. The probability of finding a significant effect for at least one measure (when there is none, assuming measures are independent) would be $1-0.95^{18}\approx 0.6>>0.05$. – Nameless Jun 25 '13 at 14:56
• @Nameless I'm pretty sure that your interpretation of a 95%-confidence interval is wrong. In a frequentist framework, the value is fixed and the endpoints of the confidence interval are random variables. For a single sample, the confidence interval either includes the true value or not. To say a single confidence interval includes the true value with a probability is not applicable. See here and here. – COOLSerdash Jun 25 '13 at 17:47
• Yes, please replace "probability" with "percent of cases" above. Which, as it happens, is the same in a frequentist framework.. – Nameless Jun 25 '13 at 19:36

So lets imagine there are only 2 symptoms and I got the following results:

### Fever: ###

Overall p-value
Wald test:
----------

Chi-squared test:
X2 = 16.8, df = 3, P(> X2) = 0.00078

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)                   1.5849     0.1103  14.366  < 2e-16 ***
relevel(agegp, "15-64")0-4    0.6323     0.3361   1.881  0.05994 .
relevel(agegp, "15-64")5-14   0.7273     0.2182   3.333  0.00086 ***
relevel(agegp, "15-64")65+   -1.0253     0.6364  -1.611  0.10718

### Cough ###

Overall p-value
Wald test:
----------

Chi-squared test:
X2 = 8.5, df = 3, P(> X2) = 0.036

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)                   2.2693     0.1416  16.024  < 2e-16 ***
relevel(agegp, "15-64")0-4   -0.3234     0.3189  -1.014  0.31051
relevel(agegp, "15-64")5-14  -0.5850     0.2050  -2.854  0.00432 **
relevel(agegp, "15-64")65+   -0.7652     0.7945  -0.963  0.33544

What would be my adjusted p-values (using Holm's method) in this scenario of having only 2 symptoms and how would I calculate confidence intervals?


text:
{{ message }}

# surban / TensorAlgDiff

Automatic Differentiation for Tensor Algebras

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# Automatic Differentiation for Tensor Algebras

Read the corresponding technical report for more details and examples.

This code computes expressions for the derivatives of element-wise defined tensor-valued functions. It can handle arguments inside the functions that are indexed by arbitrary linear combinations of the function indices. Furthermore, the function may contain (nested) sums with arbitrary ranges (even linearly depending on other indices). An example is the matrix-valued function f(a,b,c,d), which is element-wise defined by the expression

f[i; j] = exp (-sum{k}_0^4 (((a[i; k] + b[j; k]) ** 2 * c[i; i] + d[i + k] ** 3)))

Deriving derivative expressions when the mapping between function indices and argument indices is not 1:1 requires special attention. For example, for the function f_{ij} (x) = x_i^2, the derivative of some loss l=l(f(x)) w.r.t. x is (dx)_i = dl / dx_i = \sum_j (df)_{ij} 2 x_i; the sum is necessary because index j does not appear in the indices of f. Another example is f_i (x) = x_{ii}^2, where x is a matrix; here we have (dx)_{ij} = \kronecker_{i=j} (df)_i 2 x_{ii}; the Kronecker delta is necessary because the derivative is zero for off-diagonal elements. Another indexing scheme is used by f_{ij} (x) = exp x_{i+j}; here the correct derivative is (dx)_k = \sum_i (df)_{i,k-i} \exp x_k, where the range of the sum must be chosen appropriately.

Our algorithm can handle any case in which the indices of an argument are an arbitrary linear combination of the indices of the function, thus all of the above examples can be handled. Sums (and their ranges) and Kronecker deltas are automatically inserted into the derivatives as necessary. Additionally, the indices are transformed, if required (as in the last example). The algorithm outputs a symbolic expression that can be subsequently fed into a tensor algebra compiler.

For the above expression the algorithm outputs:

Derivative of f wrt. a: da[da_0; da_1] = sum{da_z0}_0^3 (((-(df[da_0; da_z0] * exp (-sum{k}_0^4 (((a[da_0; k] + b[da_z0; k]) ** 2 * c[da_0; da_0] + d[da_0 + k] ** 3))))) * c[da_0; da_0] * 2 * (a[da_0; da_1] + b[da_z0; da_1]) ** (2 - 1)))
Derivative of f wrt. b: db[db_0; db_1] = sum{db_z0}_0^2 (((-(df[db_z0; db_0] * exp (-sum{k}_0^4 (((a[db_z0; k] + b[db_0; k]) ** 2 * c[db_z0; db_z0] + d[db_z0 + k] ** 3))))) * c[db_z0; db_z0] * 2 * (a[db_z0; db_1] + b[db_0; db_1]) ** (2 - 1)))
Derivative of f wrt. c: dc[dc_0; dc_1] = if {dc_0 + -dc_1 = 0} then (sum{dc_z1}_0^4 (sum{dc_z0}_0^3 (((a[dc_1; dc_z1] + b[dc_z0; dc_z1]) ** 2 * (-(df[dc_1; dc_z0] * exp (-sum{k}_0^4 (((a[dc_1; k] + b[dc_z0; k]) ** 2 * c[dc_1; dc_1] + d[dc_1 + k] ** 3))))))))) else (0)
Derivative of f wrt. d: dd[dd_0] = sum{dd_z1}_(max [0; -2 + dd_0])^(min [4; dd_0]) (sum{dd_z0}_0^3 (((-(df[dd_0 + -dd_z1; dd_z0] * exp (-sum{k}_0^4 (((a[dd_0 + -dd_z1; k] + b[dd_z0; k]) ** 2 * c[dd_0 + -dd_z1; dd_0 + -dd_z1] + d[dd_0 + -dd_z1 + k] ** 3))))) * 3 * d[dd_0] ** (3 - 1))))

Internally, the derivatives are stored as computational trees to avoid repeated computations and thus expression blowup that otherwise occurs in symbolic differentiation. This work can easily be employed in system that generate C++ or CUDA code for expressions or be combined with a Tensor Algebra Compiler like http://tensor-compiler.org.

## Running

1. Install .NET Core 2.0 from https://www.microsoft.com/net/learn/get-started/ (packages available for all operating systems). We tested our code on Ubuntu Linux.

2. To build and run the demo execute dotnet run

3. To run the numeric verification tests run dotnet test (takes approx. 2 minutes)

## Reference

When using this work or the provided code please refer to the following publication.

Sebastian Urban, Patrick van der Smagt. Automatic Differentiation for Tensor Algebras. arXiv:1711.01348 [cs.SC], 2017.

Note that we employ some algorithms implemented in our open-source Tensor library; their source is at https://github.com/DeepMLNet/DeepNet/tree/master/Tensor. Documentation is available at https://www.deepml.net/Tensor.

Automatic Differentiation for Tensor Algebras

## Releases

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## Packages 0

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text:
File: webeq-in.html

<html>
<title>Math on the Web Example</title>
<body bgcolor=#ffffff>
Use the method of separation of variables to show that the
1-D heat conduction problem: find $u = u(x,t)$ so that
$\frac{\partial u}{\partial t}(x,t) - \alpha \frac{{\partial}^2 u}{\partial x^2}(x,t) = 0,\thicksp 0 < x < l,\thicksp t \geq 0$
$u(0,t) = 0,\thicksp t \geq 0$
$u(l,t) = 0,\thicksp t \geq 0$
$u(x,0) = f(x),\thicksp 0 \leq x \leq l$
has the solution
$u(x,t) = \sum_{n=1}^{\infty} c_n \sin \frac{n \pi x}{l} \exp \left[ -\alpha \left( \frac{n \pi}{l} \right)^2 t \right]$
where
$c_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n \pi x}{l} dx.$
</body>
</html>


text:
# Modular LaTeX document preambles and class files

I have been debating whether or not to ask this question as I don't intend to create an overly broad unanswerable, or solely opinion based question, but I have decided that there are probably some techniques that will adequately answer this question.

I am looking for approaches I can use to organize my code to simplify or clean up my class file or document preamble. Below is some example snippets and organization from my cls file. I have tried to keep as much of this customization and configuration in the cls file rather than in a complex document preamble as I want the cls file to be a template that requrires minimal customizations in constantly changing main files that would contain a preamble.

Because it's all in my cls file (which I fear is on its way to an unmanageable length). When I branch off to a new project I would like to be able start from the same class file, and just prune out and modify what is relevant. One of the primary motivations is to prevent duplication when some global concepts always need to be updated, while other customizations definitely need to be stand alone. Since \input{} seems to very nicely let me use ../.. to navigate to where I would prefer to store individual config files (except across drives)

\input{../src/private/my-newcommands}
\input{../src/private/config-glossary}  % Glossaries must be loaded before amsmath

so using this concept can the cls file be modularized and compartmentalized?

I had created some (what I had been describing as) config files that contained specific package loads and modifications for common tasks such as math, fonts, header/footer, titles, bibliography, formatting modifications, etc. But transporting some variable definitions, commands, environments, and packages (such as hyperref, cleveref, amsmath, etc) don’t seem to work when sorted into discrete tex files that are called using input or include is inside the cls file.

I can't provide a MWE for this as it seems irreproducible depending on what I put in these external tex files. For example, the color definitions and includes worked fine when grabbed with \input{} Should these by sty files instead for this to be possible?

Because a tex installation on linux has different rules about where a cls or sty file can be placed to be read, I will detail that this question is being asked with MikTex on windows in mind, but I am certainly open to comments about parallel organization structures on linux. For interests sake, one of the reasons that my cls file gets so complicated is my documents must be compatible with both xelatex and tex4ht.

Organizational example - I can't guarantee that any of this will run as I have not pruned this file with this intention, just tried to include some examples of content that I keep in my cls file:

    %_________________________________________________________________________________________________________________________________________
%
% C L A S S   F I L E   S E T T I N G S
%_________________________________________________________________________________________________________________________________________

\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{EngBird_Template}[2013/10/01

%http://tex.stackexchange.com/questions/24542/create-list-of-all-external-files-used-by-master-latex-document
%\RequirePackage{snapshot}

%\RequirePackage{standalone}
\RequirePackage{letltxmacro}
%\RequirePackage{Sweave}
%
%_________________________________________________________________________________________________________________________________________

% TODO List
% Try an organize in small modular components
% Fix bolds in section hierarchy titles
%==========================================================================================================================================
% PACKAGES REQUIRED FOR CONFIGURING REPORT  - PAGE MARGINS AND LINE SPACING
%==========================================================================================================================================
%
left=3.25cm,right=2.54cm,top=2.54cm,
bottom=2.54cm,footskip=1.0cm
%hmargin=3cm,vmargin=3.75cm %only used to get rid of a overfull \hbox warning.
]{geometry}

\RequirePackage{parskip}
\RequirePackage{setspace}

% CONFIGURE PAGE SPACING AND HYPHEN RESTRICTION CRITERION
\pagestyle{plain} % options: empty , plain , fancy
\onehalfspacing
\righthyphenmin=6
\lefthyphenmin=6

%http://tex.stackexchange.com/questions/11707/how-to-force-output-to-a-left-or-right-page
%\cleardoublepage
\RequirePackage{emptypage}  %http://tex.stackexchange.com/questions/21629/avoiding-page-numbers-in-empty-pages
\newcommand*{\cleartoleftpage}{%
\clearpage
\if@twoside
\ifodd\c@page
\hbox{}\newpage
\if@twocolumn
\hbox{}\newpage
\fi
\fi
\fi
}
%
%_________________________________________________________________________________________________________________________________________

%=========================================================================================================================================
% PACKAGES REQUIRED CREATING CUSTOM COLORS
%=========================================================================================================================================

\ifdefined\HCode
%   \RequirePackage[html]{tex4ht}
\RequirePackage[dvips]{xcolor}
\else
\RequirePackage{xcolor}%color %\PassOptionsToPackage{dvipsnames}{xcolor}
\fi

\definecolor{Crimson}{rgb}{0.6471, 0.1098, 0.1882}
\definecolor{chaptergrey}{rgb}{0.6,0,0}
\definecolor{halfgray}{gray}{0.55} % chapter numbers will be semi transparent .5 .55 .6 .0
\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}
\definecolor{Maroon}{cmyk}{0, 0.87, 0.68, 0.32}
\definecolor{RoyalBlue}{cmyk}{1, 0.50, 0, 0}
\definecolor{Black}{cmyk}{0, 0, 0, 0}
%_________________________________________________________________________________________________________________________________________
%=========================================================================================================================================
%   SNIPPETS FROM http://zoonek.free.fr/LaTeX/ TO IMPROVE FORMATTING AND APPEARENCE
%   http://tex.stackexchange.com/questions/73421/how-to-create-specific-chapter-style-in-book-documentclass
%=========================================================================================================================================

%http://zoonek.free.fr/LaTeX/LaTeX_samples_chapter/0.html
\def\thickhrulefill{\leavevmode \leaders \hrule height 1ex  \hfill \kern \z@}

%\vspace*{50\p@}%
%\vspace*{6\p@}%
{\parindent \z@ \centering \reset@font
\singlespacing
\scshape \@chapapp{} \thechapter
\par\nobreak
\vspace*{6\p@}%
\interlinepenalty\@M
\hrule
\vspace*{6\p@}%
%\fontsize{19}{6}
\Large
\bfseries #1\par\nobreak
\par
\vspace*{6\p@}%
\hrule
\vskip 20\p@
%\vskip 100\p@
}}

%\vspace*{50\p@}%
% \vspace*{6\p@}%
{\parindent \z@ \centering \reset@font
\singlespacing
\thickhrulefill
\par\nobreak
\vspace*{6\p@}%
\interlinepenalty\@M
\hrule
\vspace*{6\p@}%
%\fontsize{19}{6}
\Large
\bfseries #1\par\nobreak
\par
\vspace*{6\p@}%
\hrule
\vskip 20\p@
%\vskip 100\p@
}}

% some definitions
\def\year#1{\gdef\@year{#1}}                    % specified in coverpage.tex
\def\month#1{\gdef\@month{#1}}              % specified in coverpage.tex

\newcommand{\maketitlepage}
{
\singlespacing
\thispagestyle{empty}
\vspace*{\fill} \vspace{2cm} \begin{center}
\Huge \textcolor{Crimson}{\textit{\MyTitle}} \normalsize \\ \sc \vspace{2cm}
\vspace*{1.0cm}
\normalsize
by \\
\vspace*{1.0cm}
\large {\textit{\@author}} \normalsize \\ \sc \vspace{1cm}

\normalsize \vspace*{2.0cm}

etc \\

\end{center} \vspace*{\fill}
}
%=========================================================================================================================================

%=========================================================================================================================================
% Special einvironment modifications
%=========================================================================================================================================

%http://tex.stackexchange.com/questions/17887/prevent-abstract-environment-to-reset-pagenumbering
% either use notitlepage in the \LoadClass or this
\renewenvironment{abstract}{%
\if@twocolumn
\section*{\abstractname}%
\else
%\small
\begin{center}%
{\bfseries \abstractname\vspace{-.5em}\vspace{\z@}}%
\end{center}%
\quotation
\fi}
{\if@twocolumn\else\endquotation\fi}

%_________________________________________________________________________________________________________________________________________

%=========================================================================================================================================
%  E Q U A T I O N  C A P T I O N  S E T U P
%=========================================================================================================================================

\DeclareCaptionType{mycapequ}[][List of equations]
\captionsetup[mycapequ]{labelformat=empty}

This class file is loaded in by the main latex document setup as follows:

%=============================================================================
%   P R E A M B L E
%=============================================================================
%______________________________________________________________________________________
%
% Setup the document class and enable a package to read preamble and files externally    %______________________________________________________________________________________
\documentclass{EngBird_Template}

%----------------------------------------------------------------------
% Create a listing in the log of all files needed to process this document
\listfiles
\synctex=1 % turn synctex on automatically to sync between pdf viewer
%\setupbackend[export=yes]  % should be using with ConTeXt
%\setupbackend     [export=example.xml, css=example.css]
%----------------------------------------------------------------------

\input{../src/private/my-newcommands}
\input{../src/private/config-glossary}  % Glossaries must be loaded before amsmath

\renewcommand{\bibname}{References}   %\nocite{*}       % list all refs in database, cited or not

\begin{document}
\ifdefined\HCode    \renewcommand{\textregistered}{\special{t4ht@+\string&{35}xAE{59}}x}    \else   \fi
\input{../src/logical_contents}
\end{document}

Update

I think it might be worth mentioning that the modular spirit of this question could be applied to extend to workarounds for the texmf directory structure in an age of network computing. People like me use numerous computers and collaborate with others and as such absolute path names don't always work out great, and relative paths can be equally problematic. As a LaTeX user and enthusiast, but certainly not a developer, I find that my own code for customizations (in class and sty files) are not easily made portable, but also accessible to multiple projects, on multiple platforms, for multiple users.

• Using \input should work. Don't use \include in a .cls file. What doesn't work with \input? – cfr Jan 26 '15 at 3:45
• Related: Classes and packages – what's the difference? -- from that, I'd say anything that's not specific to a particular type of document (thesis, tech report, etc.) should be in a package. For me, \include is only for chapters. If I have my packages in version control, I can check them out into my TEXMFHOME folder on any platform (e.g., C:/users/me/texmf on Windows, /home/me/texmf on Linux). No idea what to tell you about xelatex and tex4ht. – Mike Renfro Jan 26 '15 at 3:50
• @cfr the first time I noticed my document stopped working when I compartmentalized it was with an initilization of def \newcommand{\maketitlepage}. The variable assignment and object creation are in their own tex file (and this document is called with input within begin{document}, but when I put the definition in it's own file it stopped working. – EngBIRD Jan 26 '15 at 3:58
• @MikeRenfro Sorry about the ambiguity, xelatex and tex4ht are not causing any problems so maybe I shouldn't have mentioned this, its just that having to configure with an \ifdefined\HCode and multiple instances of the same packages or package alternatives depending on the compile system really bloats the file, and would be far more manageable if I could have a more modular structure. – EngBIRD Jan 26 '15 at 4:03
• I don't understand what you mean about calling it 'within \begin{document}'. Do you \input the file in the preamble or in the document? Not that it should matter. More precisely, it should make no difference provided you input it at the point you'd otherwise include the commands. – cfr Jan 26 '15 at 23:52

I'm not sure I fully understood the setup of your class file, the ways you load various packages, and the ways new macros are defined (or existing ones are redefined). For sure, don't load packages -- typically in files with extension .sty -- via an \input statement. Instead, use \usepackage directives. If for no other reason, you should do so because \usepackage offers some protection against loading the same package twice (or more frequently...) with incompatible options. Similarly, you may want to rename the files you currently load via \input statements to have extension .sty, so that they can (should?) be loaded via \usepackage statements.


text:
# Properties of a discrete random variable

My stats course just taught me that a discrete random variable has a finite number of options ... I hadn't realized that. I would have thought, like a set of integers, it could be infinite. Googling and checking a several web pages, including a few from university courses, has failed to specifically confirm this; most sites do however say discrete random variables are countable - I suppose that means finitely numbered?

It is clear that continuous random variables are infinite even if (most?) often bounded.

But if discrete random variables have finite possibilities, what then is an infinite distribution of integers? It's neither discrete nor continuous? Is the question moot because variables either tend to be continuous & (by definition) infinite or discontinuous & finite?

• You should ask your stats course about geometric and poisson random variables Jun 13 '18 at 12:26
• It's online, so limited feedback. You're suggesting they're a third (and fourth?) type of variable, rather than just(!) distributions? Jun 13 '18 at 12:40
• A distribution is not a random variable--and ignoring that distinction has confused many. A beautiful theorem of early 20th century mathematics, the Lebesgue decomposition theorem, shows how to conceive of all distribution functions as comprised of three distinct kinds: "continuous" (which are further subdivided into absolutely continuous and continuous but not a.c.) and "discrete."
– whuber
Jun 13 '18 at 13:09
• not a good course you're taking i'm afraid Jun 13 '18 at 14:04
• To all the responses here, thank you (though some are over my head I will confess). I should probably refer to what triggered this question as on reviewing it I may have interpreted it incorrectly: a true/false question stating "A discrete random variable can take a finite number of distinct values" is considered true; with the explanation that the statement "is one of the key properties of a discrete random variable". If we surveyed farmers asking how many cattle they owned, it would be impossible to bound the number beforehand, it is theoretically infinite but discrete ... ? Jun 14 '18 at 10:46

If that's what your course said, it's wrong.

While discrete distributions can have a finite number of possible outcomes, they are not required to; you can have a discrete distribution that has an infinite number of possible outcomes - the number of elements should be no more than countable.

A common example would be a geometric distribution; consider the number of tosses of a fair coin until you get a head. There's no finite upper bound on the number of tosses that may be needed. It may take 1 toss, or 2, or 3, or 100, or any other number.

A discrete distribution could be negative (consider the difference between two such geometrically-distributed random variables; it can be any positive or negative integer).

A discrete distribution doesn't have to be over the integers, though, like in my example. That's just a common situation, not a requirement.

• So what's the actual condition that makes a distribution "discrete"? : ) Jun 13 '18 at 18:42
• The condition is that it has Lebesgue measure zero, is it not, @matthewDrury?. Which in turn is equivalent to the distribution summing to one on at most a countable set. Jun 13 '18 at 20:39
• I must admit I don't know the canonical definitions. I'm curious as to the role of accumulation points in all this. Jun 13 '18 at 20:45
• @Therkel I would think that a distribution over the Cantor Set would not be considered "discrete". Jun 13 '18 at 20:59
• After checking en.wikipedia.org/wiki/Countable_set I'm happy to accept this as the answer; the geometric distribution example is clear, and it seems to represent the consensus of responses contributed so far. Jun 14 '18 at 22:40

I'm writing an answer, with the perspective that I have only a very naive understanding of measure-theoretic probability (so, experts, please correct me!).

A (real-valued) random variable is a function $X: S \to \mathbb{R}$, where $S$ is a sample space.

$X$ is discrete if $X(S)$, the image of $S$ induced by $X$, is countable. $X$ is continuous if $X$ has an absolutely continuous CDF. (I don't know that much about absolutely continuous functions, so I can't elaborate on this point.)

However, not all random variables are only discrete or continuous. There are "mixed" random variables, where $X(s)$ has a CDF which is the sum of a step function and a continuous function with indicators.

You can also have random variables which are neither discrete nor continuous, such as the Cantor distribution.

• You actually know quite a lot about absolutely continuous distributions, because (almost by definition) an absolutely continuous distribution is one that has a density. There are continuous distributions that do not have densities: the archetypal example is the distribution induced by the Cantor function.
– whuber
Jun 13 '18 at 14:30
• If the countable image has an accumulation point, would we still say its discrete? Jun 13 '18 at 18:43
• @Matthew Yes. The example I mentioned in another comment (stats.stackexchange.com/a/104018/919), which clearly is discrete (every one of a countable number of values has a nonzero probability, so its distribution function consists of nothing but jumps) has the entire interval $[0,1]$ for its set of accumulation points.
– whuber
Jun 14 '18 at 1:52

If it [the variable] can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the variable is continuous in that interval

Therefore, a discrete random variable does not have to have a 'finite number of options', but there needs to be a non-infinitesimal gap between the possible values. This is the case with a distribution of integers, since the 'distance' between two neighboring integers is 1 and cannot be less than this. Therefore the variable is not continuous as it does not 'continue' within these gaps.

Edit: I know that there's probably better and/or more precise ways of answering this, but this is what helped me personally understand the difference.

• Just fyi, for intuition the Wikipedia characterization might be OK, but for most other purposes it's not correct. One important aspect of "continuous random variable" that it omits (out of several) is that it depends on the probabilities of its values, not just on the set of values it can attain. Your characterization of "non-infinitesimal" gap unfortunately is incorrect. I give a counterexample at stats.stackexchange.com/a/104018/919 showing a discrete variable that assigns positive probabilities to all rational numbers between $0$ and $1.$
– whuber
Jun 13 '18 at 13:05
• Some authors have said that values that get arbitrarily close together are not discrete, but I must admit I find it odd (though perhaps I am missing something). An example is the distribution of the difference of square roots of two Poisson random variates (w.real applications: people do sometimes take square roots with variables thought to be Poisson to stabilize the variance and may be interested in whether pair-differences are centered at zero). Values may be arbitrarily close together with such a variate but they are always distinct (you can enumerate each one), ...ctd Jun 13 '18 at 23:04
• ctd... and all the values it does take have positive probability. A simpler example would be the reciprocal $Y=1/X$ of a geometric random variate $X$ (number of trials form): for any $ε>0$ there are values closer together than that - so $X$ is discrete but $Y$ is not? Does anyone know of a good reason why this particular distinction (that values cannot be arbitrarily close together for it to be discrete) is drawn by some authors? Compare with the example of a discrete set ...ctd Jun 13 '18 at 23:04
• @Glen Those authors appear to confuse two different concepts of "discrete": one is the measure-theoretic idea discussed here and the other is the topological concept in which each element of a discrete set $A$ in a topological space is contained within an open set having no other elements of $A$ in it. Although it's nice that a probability measure supported on any discrete subset of the real line will be discrete, the converse is not true: discrete measures need not be supported on discrete subspaces.
– whuber
Jun 14 '18 at 1:58
• I suppose that's a mix up I had in my head. I'm a trained topologist, so discrete definitely rings in the topological context when I hear it. Thanks for clarifying @whuber. Jun 14 '18 at 2:31


text:
# Category:Relations

Relations are a very general concept; they model relationships between objects. You can think of a relation as a table (potentially with multiple columns). Formally, we have the following definition:

Definition: Relation
A relation on sets is a subset of .

## Pages in category "Relations"

The following 29 pages are in this category, out of 29 total.


text:
# Renaissance Re reveals surge in profit

Bermudian (re)insurer Renaissance Re more than doubled its annual profit in 2002 to $364.8m, from$164.4m in 2001. The return on equity was 29%.


text:
# Introduction To Econometrics: Part II- Violations of OLS Assumptions & Methods for Fixing them

Regression is the primary tool used in econometrics to infer relationships between a group of explanatory variables, X and a dependent variable, y. My previous post focused on the mechanics of Ordinary Least Squares (OLS) Regression and outlined key assumptions that, if true, make OLS estimates the Best Linear Unbiased Estimator (BLUE) for the coefficients in the regression:

$y = \beta X+\epsilon$

This post will discuss three common violations of OLS assumptions, and explain tools that have been developed for dealing with these violations. We’ll start with a violation of the assumption of a linear relationship between X and y, then discuss heteroskedasticity in the error terms and the issue of endogeniety.

### Linearity

If the relationship between X and y is not linear, OLS can no longer be used to estimate beta. A nonlinear regression of y on X has the form:

$y = g(X\beta)+\epsilon$

Where  g(X\beta) is the functional form of the nonlinear relationship between X and y and epsilon is the error term. Beta can be estimated using Nonlinear Least Squares regression (NLS). Similar  to OLS regression, NLS seeks to minimize the sum of the square error term.

$\hat{\beta} = argmin(\beta) \epsilon'\epsilon = (y-g(x\beta))^2$

To solve for beta, we again take the derivative and set it equal to zero, but for the nonlinear system there is no closed form solution, so the estimators have to be found using numerical optimization techniques.

The variance of a NLS estimator is:

$\hat{Var}_{\hat{\beta}_{NLS}} = \hat{\sigma^2}(\hat{G}'\hat{G})^{-1}$

Where G is a matrix of partial derivatives of g with respect to each Beta.

Modern numerical optimization techniques can solve many NLS equations quite easily making NLS a common alternative to OLS regression especially when there is a hypothesized functional form for the relationship between X and y.

### Heteroskedasticity

Heteroskedasticity arises within a data set when the errors do not have a constant variance with respect to X. In equation form, under heteroskedasticity:

$E(\epsilon_i^2|X ) \neq \sigma^2$

The presence of heteroskedasticity  increases the variance of Beta estimators found using OLS regression, reducing the efficiency of the estimator and causing it to no longer be the BLUE. As put by Allison (2012), OLS on heteroskedastic data puts “equal weight on all observations when, in fact, observations with larger disturbances contain less information”.

To fix this problem, econometric literature provides two options which both use a form of weighting to correct for differences in variance amongst the error terms:

1. Use the OLS estimate for beta, but calculate the variance of beta with a robust variance-covariance matrix .
2. Estimate Beta using Feasible Generalized Least Squares (FGLS)

Let’s begin with the first strategy, using OLS beta estimates with a robust variance-covariance matrix. The robust variance-covariance matrix can be derived using the Generalized Method of Moments (GMM) for the sake of brevity, I’ll omit the derivation here and skip to the final result:

$\hat{var}(\hat{\beta}) = (X'X)^{-1}(X'\hat{D}X)(X'X)^{-1}$

Where $\hat{D}$ is a matrix of square residuals from the OLS regression:

The second strategy, estimation using FGLS, requires a more involved process for estimating beta. FGLS can be accomplished through 3 steps:

1. Use OLS to find OLS estimate for beta and calculate the residuals:

$\hat{\epsilon}_i = y_i-x_i \hat{\beta}_{OLS}$

2. Regress the error term on a subset of X, which we will call Z, to get an estimate of a new parameter, theta (denoted with a tilde, but wordpress makes it difficult for me to add this in the middle of a paragraph). We then use this parameter to estimate the variance of the error term, sigma squared,  for each observation:

$\hat{\sigma}^2_i = z_i\tilde{\theta}$

A diagonal matrix, D (different than the D used for the robust variance-covariance matrix), is then constructed using these variance estimates.

3. Finally, we use the matrix D to find our FGLS estimator for beta:

$\hat{\beta}_{FGLS} = (X'\hat{D}^{-1}X)^{-1}(X'\hat{D}^{-1}y)$

The variance of of the FGLS beta etimate is then defined as:

$\hat{var}(\hat{\beta}_{FGLS} = (X'\hat{D}X)^{-1}$

### Endogeneity

Endogeneity arises when explanatory variables are correlated with the error term in a regression. This may be a result of simultaneity, when errors and explanatory variables are effected by the same exogenous influences, omitted variable bias,  when an important variable is left out of a regression, causing the over- or underestimation of the effect of other explanatory variables and the error term, measurement error or a lag in the dependent variable. Endogeniety can be hard to detect and may cause regression large errors in regression results.

A common way of correcting for endogeniety is through Instrumental Variables (IVs). Instrumental variables are explanatory variables that are highly correlated with variables that cause endogeniety but are exogenous to the system. Examples include using proximity to cardiac care centers as an IV for heart surgery when modeling health or state cigarette taxes as an IV for maternal smoking rate when modeling infant birth weight (Angrist and Kruger, 2001). For an expansive but accessible overview of IVs and their many applications, see Angrist and Kruger (2001).

A common technique for conducting a regression using IVs is 2 Stage Least Squares (2SLS) regression. The two stages of 2SLS are as follows:

1. Define Z as a new set of explanatory variables, which omits the endogenous variables and includes the IVs (which are usually not included in the original OLS regression).
2. Project Z onto the column space of X.
3. Estimate the 2SLS using this projection:

$\hat{\beta}_{2SLS} = [X'Z(Z'Z)^{-1}Z'X]^{-1}[X'Z(Z'Z)^{-1}Z'y]$

Using 2SLS regression to correct for endogeneity is fairly simple, however identifying good IVs for an endogenous variable can be extremely difficult. Finding a good IV (or set of IVs) can be enough to get one published in an economics journal (at least that’s what my economist friend told me).

## Concluding thoughts

These two posts have constituted an extremely brief introduction to the field of econometrics meant for engineers who may be interested in learning about common empirical tools employed by economists. We covered the above methods in much more detail in class and also covered other topics such as panel data, Generalize Method Of Moments estimation, Maximum Likelihood Estimation, systems of equations in regression and discrete choice modeling. Overall, I found the course (AEM 7100) to be a useful introduction to a field that I hope to learn more about over the course of my PhD.

### References:

Allison, Paul D. (2012). “Multiple regression: a primer. Pine Forge. Thousand Oaks, CA: Press Print.

Angrist, J.; Krueger, A. (2001). “Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments”. Journal of Economic Perspectives. 15 (4): 69–85. doi:10.1257/jep.15.4.69.

# An Introduction To Econometrics: Part 1- Ordinary Least Squares Regression

I took a PhD level econometrics course this semester in the Applied Economics and Management department here at Cornell and I thought I’d share some of what I learned. Overall, I enjoyed the course and learned a great deal. It was very math and theory heavy, but the Professor Shanjun Li did a nice job keeping the class lively and interesting. I would recommend the class to future EWRS students who may be looking for some grounding in econometrics, provided they’ve taken some basic statics and linear algebra courses.

So lets start with the basics, what does the term “econometrics” even mean? Hansen (2010) defined econometrics as “the unified study of economic models, mathematical statistics and economic data”. After taking this introductory course, I’m inclined to add my own definition: econometrics is “a study of the problems with regression using Ordinary Least Squares (OLS) and how to solve them”. This is obviously a gross oversimplification of the field, however, regression through OLS was the primary tool used for finding insights and patterns within data, and we spent the vast majority of the course examining it. In this post I’ll briefly summarize OLS mechanics and classical OLS assumptions. In my next post, I’ll detail methods for dealing with violations of OLS assumptions. My hope is that reading this may help you understand some key terminology and the reasoning behind why certain econometric tools are employed.

## OLS mechanics

Our primary interest when creating an econometric model is to estimate some dependent variable, y, using a observations from a set of independent variables, X. Usually y is a vector of length n, where n is the number of observations, and X is a matrix of size (n x k) where k is the number of explanatory variables (you can think of X as a table of observations, where each column contains a different variable and each row represents an observation of that variable). The goal of OLS regression is to estimate the coefficients, beta, for the model:

$y = X\beta+\epsilon$

Where beta is a k by 1 vector of coefficients on X and epsilon is a k by 1 vector of error terms.

OLS regression estimates beta by minimizing the sum of the square error term (hence the name “least squares”). Put in matrix notation, OLS estimates beta using the equation:

$\hat{\beta} = argmin_{\beta} SSE_N(\beta) = \epsilon ' \epsilon$

The optimal beta estimate can be found through the following equations:

$\epsilon = y-X\hat{\beta}$

$\epsilon ' \epsilon = (y-X\hat{\beta})'(y-X\hat{\beta})$

Taking the derivative and setting it equal to zero:

$2X'y+2X'X\hat{\beta} = 0$

Then solving for the beta estimate:

$\hat{\beta} = (X'X)^{-1}X'y$

Estimation of y using OLS regression can be visualized as the orthogonal projection of the vector y onto the column space of X. The estimated error term, epsilon, is the orthogonal distance between the projection and the true vector y.  Figure 1 shows this projection for a y that is regressed on two explanatory variables, X1 and X2.

Figure 1: OLS regression as an orthogonal projection of vector y onto the column space of matrix X. The error term, $\hat{\epsilon}$, is the orthogonal distance between y and $X\hat{\beta}$. (Image source: Wikipedia commons)

## Assumptions and properties of OLS regression

The Gauss-Markov Theorem states that under a certain set of assumptions, the OLS estimator is the Best Linear Unbiased Estimator (BLUE) for vector y.

To understand the full meaning of the Gauss-Markov theorem, it’s important to define two fundamental properties that can be used to describe estimators, consistency and efficiency. An estimator is consistent if its value will converge to the true parameter value as the number of observations goes to infinity. An estimator is efficient if its asymptotic variance is no larger than the asymptotic variance of any other possible consistent estimator for the parameter. In light of these definitions, the Gauss-Markov Theorem can be restated as: estimators found using OLS will be the most efficient consistent estimator for beta as long as the classical OLS assumptions hold. The remainder of this post will be devoted to describing the necessary assumptions for the OLS estimator to be the BLUE and detailing fixes for when these assumptions are violated.

The four classical assumptions for OLS to be the BLUE are:

1. Linearity: The relationship between X and y is linear, following the functional form:

$y = X\beta+\epsilon$.

2. Strict exogeneity: The error $\epsilon$ terms should be independent of the value of the explanatory variables, X. Put in equation form, this assumption requires:

$E(\epsilon_i|X) = 0$

$E(\epsilon_i) =0$

3.  No perfect multicollinearity: columns of X should not be correlated with each other (see my earlier post on dealing with mulitcollinearity for fixes for violations of this assumption).

4. Spherical Error: Error terms should be homoskedastic, meaning they are evenly distributed around the X values. Put in equation form:

$E(\epsilon_i^2|X) =\sigma^2$

Where $\sigma^2$ is a constant value.

$E(\epsilon_i \epsilon_j|X)=0$

Using assumption 4, we can define the variance of $\hat{\beta}$ as:

$var(\hat{\beta}_{OLS}) = \sigma^2(X'X)^{-1}$

If assumptions 1-4 hold, then the OLS estimate for beta is the BLUE, if however, any of the assumptions are broken, we must employ other methods for estimating our regression coefficients.

In my next post I’ll detail the methods econometricians use when these assumptions are violated.

### References:

Hansen, Bruce. “Econometrics”. 2010. University of Wisconsin

http://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics2010.pdf