Friday, October 17, 2014

Econometric Research Resources

The following page, put together by John Kane at the Department of Economics, SUNY-Oswego, has some very useful links for econometrics students and researchers: Econometric Research Resources


© 2014, David E. Giles

Monday, October 13, 2014

Illustrating Asymptotic Behaviour - Part III

This is the third in a sequence of posts about some basic concepts relating to large-sample asymptotics and the linear regression model. The first two posts (here and here) dealt with items 1 and 2 in the following list, and you'll find it helpful to read them before proceeding with this post:
  1. The consistency of the OLS estimator in a situation where it's known to be biased in small samples.
  2. The correct way to think about the asymptotic distribution of the OLS estimator.
  3. A comparison of the OLS estimator and another estimator, in terms of asymptotic efficiency.
Here, we're going to deal with item 3, again via a small Monte Carlo experiment, using EViews.

Nobel Prize, 2014

From the website of the Royal Swedish Academy of Sciences:

The Prize in Economic Sciences 2014

The Royal Swedish Academy of Sciences has decided to award the Sveriges Riksbanks Prize in Economic Sciences in Memory of Alfred Nobel for 2014 to Jean Tirole, Toulouse 1 Capitole University, France

“for his analysis of market power and regulation”.
Mark Thoma has an excellent round-up of related links on his blog, Economist's View.


© 2014, David E. Giles

Sunday, October 12, 2014

Illustrating Asymptotic Behaviour - Part II

This is the second in a sequence of three posts that deal with large-sample asymptotics - especially in the context of the linear regression model. The first post dealt with item 1 in this list:
  1. The consistency of the OLS estimator in a situation where it's known to be biased in small samples.
  2. The correct way to think about the asymptotic distribution of the OLS estimator.
  3. A comparison of the OLS estimator and another estimator, in terms of asymptotic efficiency.
No surprise, but this post deals with item 2. To get the most out of it, I strongly recommend reading the first post before proceeding.

Saturday, October 11, 2014

Illustrating Asymptotic Behaviour - Part I

Learning the basics about the (large sample) asymptotic behaviour of estimators and test statistics is always a challenge. Teaching this material can be challenging too!

So, in this post and in two more to follow, I'm going to talk about a small Monte Carlo experiment that illustrates some aspects of the asymptotic behaviour of the OLS estimator. I'll focus on three things:
  1. The consistency of the OLS estimator in a situation where it's known to be biased in small samples.
  2. The correct way to think about the asymptotic distribution of the OLS estimator.
  3. A comparison of the OLS estimator and another estimator, in terms of asymptotic efficiency.


Wednesday, October 1, 2014

October Reading

October already!
  • Chauvel, C. and J. O'Quigley, 2014. Tests for comparing estimated survival functions. Biometrika, 101, 535-552. 
  • Choi, I., 2014. Unit root tests for dependent and heterogeneous micropanels. Discussion Paper No. 2014-04, Research Institute for Market Economy, Sogang University.
  • Cho, J. S. and H. White, 2014. Testing the equality of two positive-definite matrices with application to in formation matrix testing. Discussion Paper, School of Economics,Yonsei University.
  • Hansen, B. E., 2013. Model averaging, asymptotic risk, and regressor groups. Quantitative Economics, in press.
  • Miller, J. I., 2014. Simple robust tests for the specification of high-frequency predictors of a low-frequency series. Mimeo., Department of Economics, University of Missouri.
  • Owen, A. B. and P. A. Roediger, 2014. The sign of the logistic regression coefficient. American Statistician, in press.
  • Westfall, P. H., 2014. Kurtosis as peakedness, 1905-2014. R.I.P.. American Statistician, 68, 191-195.

© 2014, David E. Giles

Saturday, September 20, 2014

The (Non-) Standard Asymptotics of Dickey-Fuller Tests

One of the most widely used tests in econometrics is the (augmented) Dickey-Fuller (DF) test. We use it in the context of time series data to test the null hypothesis that a series has a unit root (i.e., it is I(1)), against the alternative hypothesis that the series is I(0), and hence stationary. If we apply the test to a first-differenced time series, then the null is that the series is I(2), and the alternative hypothesis is that it is I(1), and so on.


Suppose that the time series in question is {Yt; t = 1, 2, 3, ......, T}. The so-called "Dickey-Fuller regression" is a least squares regression of the form:

                           ΔYt = [α + β t] + γYt-1 + [Σ δj ΔYt-j] + εt   .                 (1)

Here, terms in square brackets are optional; and of these the "p" ΔYt-j terms are the "augmentation terms", whose role is to ensure that the there is no autocorrelation in the equation's residuals.

Standard econometrics packages allow for three versions of (1):
  • No drift - no trend: that is, the (α + β t) terms are omitted.
  • Drift - no trend: the intercept (drift term) is included, but the linear trend term is not.
  • Drift - and - trend: both of the α and (β t) terms are included.
For example, here's the dialogue box that you see when you go to apply the DF test using the EViews package:

Friday, September 19, 2014

Least Squares, Perfect Multicollinearity, & Estimable Functions

This post is essentially an extension of another recent post on this blog. I'll assume that you've read that post, where I discussed the problem of solving linear equations of the form Ax = y, when the matrix A is singular.

Let's look at how this problem might arise in the context of estimating the coefficients of a linear regression model, y = Xβ + ε. In the previous post, I said:
"Least squares estimation leads to the so-called "normal equations":
                                 
                         X'Xb = X'y  .                                                                (1)

If the regressor matrix, X, has k columns, then (1) is a set of k linear equations in the k unknown elements of β. You'll recall that if X has full column rank, k, then (X'X) also has full rank, k, and so (X'X)-1 is well-defined. We then pre-multiply each side of (1) by (X'X)-1, yielding the familiar least squares estimator for β, namely b = (X'X)-1X'y.
So, as long as we don't have "perfect multicollinearity" among the regressors (the columns of X), we can solve (1), and the least squares estimator is defined. More specifically, a unique estimator for each individual element of β is defined.
What if there is perfect multicollinearity, so that the rank of X, and of (X'X), is less than k? In that case, we can't compute (X'X)-1, we can't solve the normal equations in the usual way, and we can't get a unique estimator for the (full) β vector."
I promised that I'd come back to the statement, "we can't get a unique estimator for the (full) β vector". Now's the time to do that.

Thursday, September 18, 2014

"Inverting" Singular Matrices

You can only invert a matrix if that matrix is non-singular. Right? Actually, that's wrong.

You see, there are various sorts of inverse matrices, and most of them apply to the situation where the original matrix is singular

Before elaborating on this, notice that this fact may be interesting in the context of estimating the coefficients of a linear regression model, y = Xβ + ε. Least squares estimation leads to the so-called "normal equations":

                                     X'Xb = X'y  .                                                                (1)

If the regressor matrix, X, has k columns, then (1) is a set of k linear equations in the k unknown elements of β. You'll recall that if X has full column rank, k, then (X'X) also has full rank, k, and so (X'X)-1 is well-defined. We then pre-multiply each side of (1) by (X'X)-1, yielding the familiar least squares estimator for β, namely b = (X'X)-1X'y.

So, as long as we don't have "perfect multicollinearity" among the regressors (the columns of X), we can solve (1), and the least squares estimator is defined. More specifically, a unique estimator for each individual element of β is defined.

What if there is perfect multicollinearity, so that the rank of X, and of (X'X), is less than k? In that case, we can't compute (X'X)-1, we can't solve the normal equations in the usual way, and we can't get a unique estimator for the (full) β vector.

Let's look carefully at the last sentence above. There are two parts of it that bear closer scrutiny:

Saturday, September 13, 2014

The Econometrics of Temporal Aggregation - IV - Cointegration

My previous post on aggregating time series data over time dealt with some of the consequences for unit roots. The next logical thing to consider is the effect of such aggregation on cointegration, and on testing for its presence.

As in the earlier discussion, we'll consider the situation where the aggregation is over "m" high-frequency periods. A lower case symbol will represent a high-frequency observation on a variable of interest; and an upper-case symbol will denote the aggregated series. So,

           Yt = yt + yt - 1 + ......+ yt - m + 1 .

If we're aggregating quarterly (flow) data to annual data, then m = 4. In the case of aggregation from monthly to quarterly data, m = 3, and so on.

We know, from my earlier post, that if yt is integrated of order one (i.e.,  I(1)), then so is Yt.

Suppose that we also have a second temporally aggregated series:

           Xt = xt + xt - 1 + ......+ xt - m + 1 .

Again, if xt is I(1) then Xt is also I(1). There is the possibility that xt and yt are cointegrated. If they are, is the same true for the aggregated series, Xt and Yt?