Econometric methods of analysis and forecasting of financial markets Lecture 4. Cointegration

This lecture helps to understand: Why the problem of non-stationarity is important in finance How to solve this problem What is cointegration How to test for cointegration

Contents Non-stationarity Cointegration. The basic idea and definitions Examples How to test for cointegration Error correction Examples of cointegration in finance

Non-stationarity Why it is important and why we should analyze it in a different way from stationary data: Stationarity and non-stationarity can affect the properties of the series Shocks Spurious regressions In case of non-stationarity the standard assumptions for asymptotic analysis will not be valid.

Non-stationarity

The basic idea and definitions The basic idea of cointegration is to test if two non-stationary variables give in result stationary variable If such combination exists, these variables are said to be cointegrated

The basic idea and definitions

Examples

Money demand models: cointegration between money, nominal income, prices, and interest rates. Growth theory models: cointegration between income, consumption, and investment. The Fisher equation: cointegration between nominal interest rates and inflation. The expectations hypothesis of the term structure: cointegration between nominal interest rates at different maturities.

The basic idea and definitions

How to test for cointegration With 2 variables intuition is simple: test in levels. If errors are I(0) and the original series were I(1) then the variables are cointegrated. With more than 2 variables: Engle and Granger methodology Johansen methodology

Error correction

Engle-Granger methodology 2 variables: x and z 4-steps procedure: 1.Test variables for the order of integration (variables should be non-stationary and integrated of the same order) 2.Estimate the long-run equilibrium relationship and test if the errors are stationary or not 3.If variables are cointegrated, estimate the Error Correction Model 4.Test the adequacy of the model (are the errors of ECM white noise, speed of adjustment, interaction among all variables)

Problems with E&G methodology In step 2 LR equilibrium regression is not unique What to do when there are more than 2 variables If you make a mistake at the first step, it is saved at the second step There is no testing procedure for cointegration at step 1

Johansen methodology Avoids the use of 2-steps estimator Allows for multiple cointegrating vectors It’s multivariate version of Dickey-Fuller test Johansen test is affected by length of lag. Better way: try with different lags

Johansen methodology

Examples in finance: PPP (Purchasing Power Parity) Cointegration between international bond markets Spot and futures prices for a given commodity or asset Ratio of relative prices and an exchange rate Equity prices and dividends.

Conclusions We’ve learned why the problem of non- stationarity is important in finance How to solve this problem What is cointegration and how to test for it

References Brooks C. Introductory Econometrics for Finance. Cambridge University Press Cuthbertson K., Nitzsche D. Quantitative Financial Economics. Wiley Tsay R.S. Analysis of Financial Time Series, Wiley, Y. Ait-Sahalia, L. P. Hansen. Handbook of Financial Econometrics: Tools and Techniques. Vol. 1, 1st Edition Alexander C. Market Models: A Guide to Financial Data Analysis. Wiley Cameron A. and Trivedi P.. Microeconometrics. Methods and Applications Lai T. L., Xing H. Statistical Models and Methods for Financial Markets. Springer Poon S-H. A practical guide for forecasting financial market volatility. Wiley, Rachev S.T. et al. Financial Econometrics: From Basics to Advanced Modeling Techniques, Wiley, 2007.