Sunday, October 4, 2015

Cointegration & Granger Causality

Today, I had a query from a reader of this blog regarding cointegration and Granger causality. 

Essentially, the email said:
"I tested two economic time-series and found them to be cointegrated. However, when I then tested for Granger  causality, there wasn't any. Am I doing something wrong?"
First of all, the facts:

  • If two time series, X and Y, are cointegrated, there must exist Granger causality either from X to Y, or from Y to X, both in both directions.
  • The presence of Granger causality in either or both directions between X and Y does not necessarily imply that the series will be cointegrated.
Now, what about the question that was raised?

Truthfully, not enough information has been supplied for anyone to give a definitive answer.
  1. What is the sample size? Even if applied properly, tests for Granger non-causality have only asymptotic validity (unless you bootstrap the test).
  2. How confident are you that the series are both I(1), and that you should be testing for cointegration in the first place?
  3. What is the frequency of the data, and have they been seasonally adjusted? This can affect the unit root tests, cointegration test, and Granger causality test.
  4. How did you test for cointegration - the Engle-Granger 2-step approach, or via Johansen's methodology?
  5. How did you test for Granger non-causality? Did you use a modified Wald test, as in the Toda-Yamamoto approach?
  6. Are there any structural breaks in either of the time-series? These ail likely any or all of the tests that you have performed.
  7. Are you sure that you correctly specified the VAR model used for the causality testing, and the VAR model on which Johansen's tests are based (if you used his methodology to test for cointegration)?
The answers to some or all of these questions will contain the key to why you obtained an apparently illogical result.

Theoretical results in econometrics rely on assumptions/conditions that have to be satisfied. If they're not, then don't be surprised by the empirical results that you obtain.

© 2015, David E. Giles

Friday, October 2, 2015

Illustrating Spurious Regressions

I've talked a bit about spurious regressions a bit in some earlier posts (here and here). I was updating an example for my time-series course the other day, and I thought that some readers might find it useful.

Let's begin by reviewing what is usually meant when we talk about a "spurious regression".

In short, it arises when we have several non-stationary time-series variables, which are not cointegrated, and we regress one of these variables on the others.

In general, the result that we get are nonsensical, and the problem is only worsened if we increase the sample size. This phenomenon was observed by Granger and Newbold (1974), and others, and Phillips (1986) developed the asymptotic theory that he then used to prove that in a spurious regression the Durbin-Watson statistic converges in probability to zero; the OLS parameter estimators and R2 converge to non-standard limiting distributions; and the t-ratios and F-statistic diverge in distribution, as T ↑ ∞ .

Let's look at some of these results associated with spurious regressions. We'll do so by means of a simple simulation experiment.

Thursday, October 1, 2015

What NOT To Do When Data Are Missing

Here's something that's very tempting, but it's not a good idea.

Suppose that we want to estimate a regression model by OLS. We have a full sample of size n for the regressors, but one of the values for our dependent variable, y, isn't available. Rather than estimate the model using just the (n - 1) available data-points, you might think that it would be preferable to use all of the available data, and impute the missing value for y.

Fair enough, but what imputation method are you going to use?

For simplicity, and without any loss of generality, suppose that the model has a single regressor,
                yi = β xi + εi ,                                                                       (1)

 and it's the nth value of y that's missing. We have values for x1, x2, ...., xn; and for y1, y2, ...., yn-1.

Here's a great idea! OLS will give us the Best Linear Predictor of y, so why don't we just estimate (1) by OLS, using the available (n - 1) sample values for x and y; use this model (and xn) to get a predicted value (y*n) for yn; and then re-estimate the model with all n data-points: x1, x2, ...., xn; y1, y2, ...., yn-1, y*n.

Unfortunately, this is actually a waste of time. Let's see why.

Wednesday, September 30, 2015

Reading List for October

Some suggestions for the coming month:

© 2015, David E. Giles

Friday, September 25, 2015

Thanks, Dan!

Quite out of the blue, Dan Getz kindly sent me a nice LaTeX version of my hand-written copy of "The Solution", given in my last post.

Dan used the ShareLaTeX site - .

So, here's a nice pdf file with The Solution.

Thanks a million, Dan - that was most thoughtful of you!

© 2015, David E. Giles

Tuesday, September 22, 2015

The Solution

You can find a solution to the problem posed in yesterday's post here.

I hope you can read my writing!

p.s.: Dan Getz kindly supplied a LaTeX version - here's the pdf file. Thanks, Dan!

© 2015, David E. Giles

Monday, September 21, 2015

Try This Problem

Here's a little exercise for you to work on:

We know from the Gauss-Markhov Theorem that within the class of linear and unbiased estimators, the OLS estimator is most efficient. Because it is unbiased, it therefore has the smallest possible Mean Squared Error (MSE) within the linear and unbiased class of estimators. However, there are many linear estimators which, although biased, have a smaller MSE than the OLS estimator. You might then think of asking: 
“Why don’t I try and find the linear estimator that has the smallest possible MSE?”
(a) Show that attempting to do this yields an “estimator” that can’t actually be used in practice.

(You can do this using the simple linear regression model without an intercept, although the result generalizes to the usual multiple linear regression model.)

(b) Now, for the simple regression model with no intercept, 

         yi = β xi + εi       ;     εi ~ i.i.d. [0 , σ2] ,

find the linear estimator, β* , that minimizes the quantity:

h[Var.(β*) / σ2] + (1 - h)[Bias(β*)/ β]2 , for 0 < h < 1.

Is  β* a legitimate estimator, in the sense that it can actually be applied in practice?

The answer will follow in a subsequent post.

© 2015, David E. Giles

Tuesday, September 1, 2015

September Reading List

  • Abeln, B. and J. P. A. M. Jacobs, 2015. Seasonal adjustment with and without revisions: A comparison of X-13ARIMA-SEATS and CAMPLET. CAMA Working Paper 25/2015, Crawford School of Public Policy, Australian National University.
  • Chan, J. C. C. and A. L. Grant, 2015. A Bayesian model comparison for trend-cycle decompositions of output. CAMA Working Paper 31/2015, Crawford School of Public Policy, Australian National University.
  • Chen, K. and K-S. Chan, 2015. A note on rank reduction in sparse multivariate regression. Journal of Statistical Theory and Practice, in press.
  • Fan, Y., S. Pastorello, and E. Renault, 2015. Maximization by parts in extremum estimation. Econometrics Journal, 18, 147-171.
  • Horowitz, J., 2014. Variable selection and estimation in high-dimensional models. Cemmap Working Paper CWP35/15, Institute of Fiscal Studies, Department of Economics, University College London.
  • Larson, W., 2015. Forecasting an aggregate in the presence of structural breaks in the disaggregates. RPF Working Paper No. 2015-002, Research Program on Forecasting, Center of Economic Research, George Washington University.

© 2015, David E. Giles

Wednesday, August 26, 2015

Biased Estimation of Marginal Effects

I began a recent post with the comment:
"One thing that a lot of practitioners seem to be unaware of (or they choose to ignore it) is that in many of the common situations where we use regression analysis to estimate elasticities, these estimators are biased.
And that's true even if all of the conditions needed for the coefficient estimator (e.g., OLS) to be unbiased are fully satisfied."
Exactly the same point can be made in respect of estimated marginal effects, and that's what this post is about.

Tuesday, August 25, 2015

The Distribution of a Ratio of Correlated Normals

Suppose that the random variables X1 and X2 are jointly distributed as bivariate Normal, with means of θ1 and θ2, variances of σ12 and σ22 respectively, and a correlation coefficient of ρ.

In this post we're going to be looking at the distribution of the ratio, W = (X1 / X2).

You probably know that if X1 and X2 are independent standard normal variables, then W follows a Cauchy distribution. This will emerge as a special case in what follows.

The more general case that we're concerned with is of interest to econometricians for several reasons.