Reevaluating probabilities based on piece of evidence

This actually has to do with the Bayes' theorem. However, in simple problems one can use a dead simple approach: just find probabilities of all elementary events. This post builds upon the post on Significance level and power of test, including the notation. Be sure to review that post.

Here is an example from the guide for Quantitative Finance by A. Patton (University of London course code FN3142).

Activity 7.2 Consider a test that has a Type I error rate of 5%, and power of 50%.

Suppose that, before running the test, the researcher thinks that both the null and the alternative are equally likely.

If the test indicates a rejection of the null hypothesis, what is the probability that the null is false?

If the test indicates a failure to reject the null hypothesis, what is the probability that the null is true?

Denote events R = {Reject null}, A = {fAil to reject null}; T = {null is True}; F = {null is False}. Then we are given:

(1)

(2)

(1) and (2) show that we can find and and therefore also and Once we know probabilities of elementary events, we can find everything about everything.

Figure 1. Elementary events

Answering the first question: just plug probabilities in

Answering the second question: just plug probabilities in

Patton uses the Bayes' theorem and the law of total probability. The solution suggested above uses only additivity of probability.

In this post we discuss several interrelated concepts: null and alternative hypotheses, type I and type II errors and their probabilities. Review the definitions of a sample space and elementary events and that of a conditional probability.

Type I and Type II errors

Regarding the true state of nature we assume two mutually exclusive possibilities: the null hypothesis (like the suspect is guilty) and alternative hypothesis (the suspect is innocent). It's up to us what to call the null and what to call the alternative. However, the statistical procedures are not symmetric: it's easier to measure the probability of rejecting the null when it is true than other involved probabilities. This is why what is desirable to prove is usually designated as the alternative.

Usually in books you can see the following table.

Decision taken

Fail to reject null

Reject null

State of nature

Null is true

Correct decision

Type I error

Null is false

Type II error

Correct decision

This table is not good enough because there is no link to probabilities. The next video does fill in the blanks.

This is a large topic which requires several posts or several book chapters. During a conference in Sweden in 2010, a Swedish statistician asked me: "What is Econometrics, anyway? What tools does it use?" I said: "Among others, it uses linear regression." He said: "But linear regression is a general statistical tool, why do they say it's a part of Econometrics?" My answer was: "Yes, it's a general tool but the name Econometrics emphasizes that the motivation for its applications lies in Economics".

Both classical assumptions and their violations should be studied with this point in mind: What is the Economics and Math behind each assumption?

Violations of the first three assumptions

We consider the simple regression

(1)

Make sure to review the assumptions. Their numbering and names sometimes are different from what Dougherty's book has. In particular, most of the time I omit the following assumption:

A6. The model is linear in parameters and correctly specified.

When it is not linear in parameters, you can think of nonlinear alternatives. Instead of saying "correctly specified" I say "true model" when a "wrong model" is available.

A1. What if the existence condition is violated? If variance of the regressor is zero, the OLS estimator does not exist. The fitted line is supposed to be vertical, and you can regress on . Violation of the existence condition in case of multiple regression leads to multicollinearity, and that's where economic considerations are important.

A2. The convenience condition is called so because when it is violated, that is, the regressor is stochastic, there are ways to deal with this problem: finite-sample theory and large-sample theory.

A3. What if the errors in (1) have means different from zero? This question can be divided in two: 1) the means of the errors are the same: for all and 2) the means are different. Read the post about centering and see if you can come up with the answer for the first question. The means may be different because of omission of a relevant variable (can you do the math?). In the absence of data on such a variable, there is nothing you can do.

In this post we looked at dependence of EARNINGS on S (years of schooling). In the end I suggested to think about possible variations of the model. Specifically, could the dependence be nonlinear? We consider two answers to this question.

Quadratic regression

This name is used for the quadratic dependence of the dependent variable on the independent variable. For our variables the dependence is

.

Note that the dependence on S is quadratic but the right-hand side is linear in the parameters, so we still are in the realm of linear regression. Video 1 shows how to run this regression.

Video 1. Running quadratic regression in Stata

Nonparametric regression

The general way to write this model is

The beauty and power of nonparametric regression consists in the fact that we don't need to specify the functional form of dependence of on . Therefore there are no parameters to interpret, there is only the fitted curve. There is also the estimated equation of the nonlinear dependence, which is too complex to consider here. I already illustrated the difference between parametric and nonparametric regression. See in Video 2 how to run nonparametric regression in Stata.

Running simple regression in Stata is, well, simple. It's just a matter of a couple of clicks. Try to make it a small research.

Obtain descriptive statistics for your data (Statistics > Summaries, tables, and tests > Summary and descriptive statistics > Summary statistics). Look at all that stuff you studied in introductory statistics: units of measurement, means, minimums, maximums, and correlations. Knowing the units of measurement will be important for interpreting regression results; correlations will predict signs of coefficients, etc. In your report, don't just mechanically repeat all those measures; try to find and discuss something interesting.

Visualize your data (Graphics > Twoway graph). On the graph you can observe outliers and discern possible nonlinearity.

After running regression, report the estimated equation. It is called a fitted line and in our case looks like this: Earnings = -13.93+2.45*S (use descriptive names and not abstract X,Y). To see if the coefficient of S is significant, look at its p-value, which is smaller than 0.001. This tells us that at all levels of significance larger than or equal to 0.001 the null that the coefficient of S is significant is rejected. This follows from the definition of p-value. Nobody cares about significance of the intercept. Report also the p-value of the F statistic. It characterizes significance of all nontrivial regressors and is important in case of multiple regression. The last statistic to report is R squared.

Think about possible variations of the model. Could the dependence of Earnings on S be nonlinear? What other determinants of Earnings would you suggest from among the variables in Dougherty's file?

Figure 1. Looking at data. For data, we use a scatterplot.

Figure 2. Running regression (Statistics > Linear models and related > Linear regression)

Introduction to Stata: Stata interface, how to use Stata Help, how to use Data Editor and how to graph data. Important details to remember:

In any program, the first thing to use is Help. I learned everything from Help and never took any programming courses.

The number of observations for all variables in one data file must be the same. This can be a problem if, for example, you want to see out-of-sample predictions.

In Data Editor, numeric variables are displayed in black and strings are displayed in red.

The name of the hidden variable that counts observations is _n

If you have several definitions of graphs in two-way graphs menu, they will be graphed together or separately, depending on what is enabled/disabled.

See details in videos. Sorry about the background noise!

Video 1. Stata interface. The windows introduced: Results, Command, Variables, Properties, Review and Viewer.

Video 2. Using Stata Help. Help can be used through the Review window or in a separate pdf viewer. Eviews Help is much easier to understand.

Video 3. Using Data Editor. How to open and view variables, the visual difference between numeric variables and string variables. The lengths of all variables in the same file must be the same.

Video 4. Graphing data. To graph a variable, you need to define its graph and then display it. It is possible to display more than one variable on the same chart.

Autoregressive–moving-average (ARMA) models were suggested in 1951 by Peter Whittle in his PhD thesis. Do you think he played with data and then came up with his model? No, he was guided by theory. The same model may describe visually very different data sets, and visualization rarely leads to model formulation.

Recall that the main idea behind autoregressive processes is to regress the variable on its own past values. In case of moving averages, we form linear combinations of elements of white noise. Combining the two ideas, we obtain the definition of the autoregressive–moving-average process:

(1)

It is denoted ARMA(p,q), where p is the number of included past values and q is the number of included past errors (AKA shocks to the system). We should expect a couple of facts to hold for this process.

If its characteristic polynomial has roots outside the unit circle, the process is stable and should be stationary.

A stable process can be represented as an infinite moving average. Such a representation is in fact used to analyze its properties.

The quantity can be called an instantaneous effect of on . This effect accumulates over time (the value at is influenced by the value at , which, in turn, is influenced by the value at and so on). Therefore the long-run interpretation of the coefficients is complicated. Comparison of Figures 1 and 2 illustrates this point.

Exercise. For the model find the mean (just modify this argument).

Figure 1. Simulated AR process

Figure 2. Simulated MA

Question. Why in (1) the current error has coefficient 1? Choose the answer you like:

1) We never used the current error with a nontrivial coefficient.

2) It is logical to assume that past shocks may have an aftereffect (measured by thetas) on the current different from 1 but the effect of the current shock should be 1.

3) Mathematically, the case when instead of we have with some nonzero can be reduced to the case when the current error has coefficient 1. Just introduce a new white noise and rewrite the model using it.

Moving average processes: this time the intuition is mathematical and even geometric.

Review and generalize

Science is a vertical structure, as I say in my book, and we have long past the point after which looking back is as important as looking forward. So here are a couple of questions for the reader to review the past material.

Q1. What is a stochastic process? (Answer: imagine a real line with a random variable attached to each integer point.)

Q2. There are good (stationary) processes and bad (all other) processes. How do you define the good ones? (Hint: Properties of means, covariances and variances are bread and butter of professionals.)

Q3. White noise is a simplest (after a constant) type of a stationary process. Give the definition, and don't hope for a hint. Do you realize that the elements of white noise don't interact with one another in the sense that covariance between each two of them is zero?

Idea. Define a class of stationary processes by forming linear combinations of elements of white noise.

Q4. A simple realization of this idea is given here. How do you generalize it?

Answer. The process

(1)

where is white noise, is called a moving average process of order and denoted MA(q).

Remarks. 1) The "moving average" name may be misleading. In Finance we use that name when the coefficients sum to one and are positive. Here the thetas do not necessarily sum to one and may change sign.

2) It would be better to say a "moving linear combination". The coefficients of the linear combination do not change but are applied to a moving segment of the white noise, starting from the element dated and going back to the element dated . In this sense we say that (1) involves the segment .

3) In Economics and Finance, the errors are treated as shocks. (1) tells us that the process is a result of the current shock and previous shocks.

Moving average properties

First stationarity condition. , should be absolutely obvious by now.

Second stationarity condition. Variance does not depend on time:

because only products have nonzero expectations.

Third stationarity condition. Here is where geometry is useful. If one linear combination involves the segment and the other - the segment , then under what condition these segments do not overlap? Answer: if the distance between the points is larger than . In this case the linear combinations do not have common elements and is zero.

Exercise. (I leave you the tedious part). Calculate for .

Autoregressive processes: going from the particular to the general is the safest option. Simple observations are the foundation of any theory.

Intuition

Figure 1. Electricity load in France and Great Britain for 2001 to 2006

If you have only one variable, what can you regress it on? Only on its own past values (future values are not available at any given moment). Figure 1 on electricity demand from a paper by J.W. Taylor illustrates this. A low value of electricity demand, say, in summer last year, will drive down its value in summer this year. Overall, we would expect the electricity demand now to depend on its values in the past 12 months. Another important observation from this example is that probably this time series is stationary.

AR(p) model

We want a definition of a class of stationary models. From this example we see that excluding the time trend increases chances of obtaining a stationary process. The idea to regress the process on its own past values is realized in

(1)

Here is some positive integer. However, both this example and the one about random walk show that some condition on the coefficients will be required for (1) to be stationary. (1) is called an autoregressive process of order and denoted AR(p).

Exercise 1. Repeat calculations on AR(1) process to see that in case for (1) the stability condition is sufficient for stationarity (that is, the coefficient has no impact on stationarity).

Question. How does this stability condition generalize to AR(p)?

Characteristic polynomial

Denote the lag operator defined by . More generally, its powers are defined by . Then (1) can be rewritten as

Whoever first did this wanted to solve the equation for . Sending all terms containing to the left we have

The identity operator is defined by , so . Factoring out we get

(2)

Finally, formally solving for we have

(3)

Definition 1. In replace the identity by 1 and powers of the lag operator by powers of a real number to obtain the definition of the characteristic polynomial:

Definition 2. We say that model (1) is stable if its characteristic polynomial (3) has roots outside the unit circle, that is, the roots are larger than 1 in absolute value.

Under this stability condition the passage from (2) to (3) can be justified. For AR(1) process this actually has been done.

Example 1. In case of a first-order process, has one root which lies outside the unit circle exactly when

Example 2. In case of a second-order process, has two roots. If both of them are larger than 1 in absolute value, then the process is stable. The formula for the roots of a quadratic equation is well-known but stating it here wouldn't add much to what we know. Most statistical packages, including Stata, have procedures for checking stability.

Remark. Hamilton uses a different definition of the characteristic polynomial (linked to vector autoregressions), that's why in his definition the roots of the characteristic equation should lie inside the unit circle.

Stationary processes: Here we consider an example that appeared in the Econometrics exams of the University of London.

Example 3. Consider the process defined by

(1)

where is white noise:

(2) , for all and for all

In Economics, the error is treated as a shock in the current period. So in (1) sustains a shock in period . There is no reason to put a coefficient in front of (or, you could say, the coefficient is 1). Besides, remembers the shock from the previous period, and the aftereffect of this shock is measured by the coefficient .

Observation. Before we proceed with the analysis of this model, it is a good idea to elucidate the techniques of working with white noise. For this, we can rewrite the conditions on the second moments of errors from (2) as

if and if

In words, the expected value of a product of any error with itself is and expected values of products of different errors disappear. This fact is used many times in Econometrics, in particular, here. I am mentioning it here because some students have problems with it even in the second semester.

First stationarity condition. Obviously,

(3) for all .

Second stationarity condition. Variance does not depend on time:

(4)

because only products have nonzero expectations.

Third stationarity condition. Using the observation above, let us look at the expectation

(5)

The error from the first parenthesis does not have a match in the second one, while the error from the second parenthesis does not have a match in the first one. Thus, (5) equals

(6)

Similarly,

(7)

Further, if the distance between points is larger than one, then in

(8)

expected values of all products will be zero. Equations (4), (6)-(8) are summarized as follows:

if , if , and if .

Recalling the first stationarity condition, we can rewrite this as

if , if , and if

Thus the covariance is indeed a function of only .

The process we studied is called a moving average of order 1 and denoted MA(1).