Today's talk: “Analysis of variance in the central limit theorem”
The talk is about results, which are a combination of methods of the function theory, functional analysis and probability theory. The intuition underlying the central limit theorem will be described, and the history and place of the results of the author in modern theory will be highlighted.
Today's talk: “Analysis of variance in the central limit theorem”
Useful facts about independence
In the one-dimensional case the economic way to define normal variables is this: define a standard normal variable and then a general normal variable as its linear transformation.
In case of many dimensions, we follow the same idea. Before doing that we state without proofs two useful facts about independence of random variables (real-valued, not vectors).
Theorem 1. Suppose variables have densities Then they are independent if and only if their joint density is a product of individual densities:
Theorem 2. If variables are normal, then they are independent if and only if they are uncorrelated:
The necessity part (independence implies uncorrelatedness) is trivial.
Let be independent standard normal variables. A standard normal variable is defined by its density, so all of have the same density. We achieve independence, according to Theorem 1, by defining their joint density to be a product of individual densities.
Definition 1. A standard normal vector of dimension is defined by
Definition 2. For a matrix and vector of compatible dimensions a normal vector is defined by
(recall that variance of a vector is always nonnegative).
Distributions derived from normal variables
In the definitions of standard distributions (chi square, t distribution and F distribution) there is no reference to any sample data. Unlike statistics, which by definition are functions of sample data, these and other standard distributions are theoretical constructs. Statistics are developed in such a way as to have a distribution equal or asymptotically equal to one of standard distributions. This allows practitioners to use tables developed for standard distributions.
Exercise 1. Prove that converges to 1 in probability.
Proof. For a standard normal we have and (both properties can be verified in Mathematica). Hence, and
Now the statement follows from the simple form of the law of large numbers.
Exercise 1 implies that for large the t distribution is close to a standard normal.
Application: estimating sigma squared
Consider multiple regression
(a) the regressors are assumed deterministic, (b) the number of regressors is smaller than the number of observations (c) the regressors are linearly independent, and (d) the errors are homoscedastic and uncorrelated,
Usually students remember that should be estimated and don't pay attention to estimation of Partly this is because does not appear in the regression and partly because the result on estimation of error variance is more complex than the result on the OLS estimator of
Definition 1. Let be the OLS estimator of . is called the fitted value and is called the residual.
Exercise 1. Using the projectors and show that and
Proof. The first equation is obvious. From the model we have Since we have further
Definition 2. The OLS estimator of is defined by
Exercise 2. Prove that is unbiased:
Proof. Using projector properties we have
Expectations of type and would be easy to find from (2). However, we need to find where there is an obstructing See how this difficulty is overcome in the next calculation.
( is a scalar, so its trace is equal to itself)
(the regressors and hence are deterministic, so we can use linearity of )
because this is the dimension of the image of Therefore Thus, and
Application: Ordinary Least Squares estimator
Generalized Pythagoras theorem
Exercise 1. Let be a projector and denote Then
Proof. By the scalar product properties
is symmetric and idempotent, so
This proves the statement.
Ordinary Least Squares (OLS) estimator derivation
Problem statement. A vector (the dependent vector) and vectors (independent vectors or regressors) are given. The OLS estimator is defined as that vector which minimizes the total sum of squares
Denoting we see that and that finding the OLS estimator means approximating with vectors from the image should be linearly independent, otherwise the solution will not be unique.
Assumption. are linearly independent. This, in particular, implies that
Exercise 2. Show that the OLS estimator is
Proof. By Exercise 1 we can use Since belongs to the image of doesn't change it: Denoting also we have
(by Exercise 1)
This shows that is a lower bound for This lower bound is achieved when the second term is made zero. From
we see that the second term is zero if satisfies (2).
Usually the above derivation is applied to the dependent vector of the form where is a random vector with mean zero. But it holds without this assumption. See also simplified derivation of the OLS estimator.
Different faces of vector variance: again visualization helps
In the previous post we defined variance of a column vector with components by
In terms of elements this is the same as:
So why knowing the structure of this matrix is so important?
Let be random variables and let be numbers. In the derivation of the variance of the slope estimator for simple regression we have to deal with the expression of type
Question 1. How do you multiply a sum by a sum? I mean, how do you use summation signs to find the product ?
Answer 1. Whenever you have problems with summation signs, try to do without them. The product
should contain ALL products Again, a matrix visualization will help:
The product we are looking for should contain all elements of this matrix. So the answer is
Formally, we can write (the sum does not depend on the index of summation, this is another point many students don't understand) and then perform the multiplication in (3).
Question 2. What is the expression for (2) in terms of covariances of components?
Answer 2. If you understand Answer 1 and know the relationship between variances and covariances, it should be clear that
Question 3. In light of (1), separate variances from covariances in (4).
Answer 3. When we have which are diagonal elements of (1). Otherwise, for we get off-diagonal elements of (1). So the answer is
Once again, in the first sum on the right we have only variances. In the second sum, the indices are assumed to run from to , excluding the diagonal
Corollary. If are uncorrelated, then the second sum in (5) disappears:
This fact has been used (with a slightly different explanation) in the derivation of the variance of the slope estimator for simple regression.
Question 4. Note that the matrix (1) is symmetric (elements above the main diagonal equal their mirror siblings below that diagonal). This means that some terms in the second sum on the right of (5) are repeated twice. If you group equal terms in (5), what do you get?
Answer 4. The idea is to write
that is, to join equal elements above and below the main diagonal in (1). For this, you need to figure out how to write a sum of the elements that are above the main diagonal. Make a bigger version of (1) (with more off-diagonal elements) to see that the elements that are above the main diagonal are listed in the sum This sum can also be written as Hence, (5) is the same as
Unlike (6), this equation is applicable when there is autocorrelation.
Significance level and power of test
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.
|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.
Significance level and power of test
The conclusion from the video is that
Violations of classical assumptions
This will be a simple post explaining the common observation that "in Economics, variability of many variables is proportional to those variables". Make sure to review the assumptions; they tend to slip from memory. We consider the simple regression
One of classical assumptions is
Homoscedasticity. All errors have the same variances: for all .
We discuss its opposite, which is
Heteroscedasticity. Not all errors have the same variance. It would be wrong to write it as for all (which means that all errors have variance different from ). You can write that not all are the same but it's better to use the verbal definition.
Remark about Video 1. The dashed lines can represent mean consumption. Then the fact that variation of a variable grows with its level becomes more obvious.
Homoscedasticity was used in the derivation of the OLS estimator variance; under heteroscedasticity that expression is no longer valid. There are other implications, which will be discussed later.
Companies example. The Samsung Galaxy Note 7 battery fires and explosions that caused two recalls cost the smartphone maker at least $5 billion. There is no way a small company could have such losses.
GDP example. The error in measuring US GDP is on the order of $200 bln, which is comparable to the Kazakhstan GDP. However, the standard deviation of the ratio error/GDP seems to be about the same across countries, if the underground economy is not too big. Often the assumption that the standard deviation of the regression error is proportional to one of regressors is plausible.
To see if the regression error is heteroscedastic, you can look at the graph of the residuals or use statistical tests.
Violations of classical assumptions
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
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.
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.
Violations of A4 and A5 will be treated later.
Nonlinear least squares
Here we explain the idea, illustrate the possible problems in Mathematica and, finally, show the implementation in Stata.
Idea: minimize RSS, as in ordinary least squares
Observations come in pairs . In case of ordinary least squares, we approximated the y's with linear functions of the parameters, possibly nonlinear in x's. Now we use a function which may be nonlinear in . We still minimize RSS which takes the form . Nonlinear least squares estimators are the values that minimize RSS. In general, it is difficult to find the formula (closed-form solution), so in practice software, such as Stata, is used for RSS minimization.
Simplified idea and problems in one-dimensional case
Suppose we want to minimize . The Newton algorithm (default in Stata) is an iterative procedure that consists of steps:
- Select the initial value .
- Find the derivative (or tangent) of RSS at . Make a small step in the descent direction (indicated by the derivative), to obtain the next value .
- Repeat Step 2, using as the starting point, until the difference between the values of the objective function at two successive points becomes small. The last point will approximate the minimizing point.
- The minimizing point may not exist.
- When it exists, it may not be unique. In general, there is no way to find out how many local minimums there are and which ones are global.
- The minimizing point depends on the initial point.
See Video 1 for illustration in the one-dimensional case.
Problems illustrated in Mathematica
Here we look at three examples of nonlinear functions, two of which are considered in Dougherty. The first one is a power functions (it can be linearized applying logs) and the second is an exponential function (it cannot be linearized). The third function gives rise to two minimums. The possibilities are illustrated in Mathematica.
Finally, implementation in Stata
Here we show how to 1) generate a random vector, 2) create a vector of initial values, and 3) program a nonlinear dependence.
Alternatives to simple regression in Stata
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.
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.
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.