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Time Series Econometrics

Time Series Econometrics. Ch7 Some Basic Concepts. Ch7 Basics. 1. Stochastic Processes 2. Stationarity Processes 3. Purely Random processes 4. Nonstationary Processes 5. Random Walk Models 6. Unit Root Tests 7. Cointegration and Cointegration Tests 8. Error Correction Mechanism

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Time Series Econometrics

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  1. Time Series Econometrics Ch7 Some Basic Concepts

  2. Ch7 Basics • 1. Stochastic Processes • 2. Stationarity Processes • 3. Purely Random processes • 4. Nonstationary Processes • 5. Random Walk Models • 6. Unit Root Tests • 7. Cointegration and Cointegration Tests • 8. Error Correction Mechanism • 9. Granger Causality Test

  3. Background • Regression analysis based on time series data implicitly assumes that the underlying time series are stationary. The classical t test, F tests, etc. are based on this assumption. • In practice most economic time series are nonstationary. (spurious/ nonsensical regression)

  4. Spurious Regression • Regression of one time series variable on one or more time series variables often can give nonsensical or spurious results. • Spurious regression often shows a significant relationship between variables, but in fact, this kind of relationship does not exist. This phenomenon is known as spurious regression. • An is a good rule of thumb to suspect that the estimated regression is spurious.

  5. 1. Stochastic Processes • A random or stochastic process is a collection of random variables ordered in time. • The term “stochastic” comes from the Greek word “stochos,” which means a target or bull’s eye.

  6. 2. Stationary Stochastic Processes • A stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on the distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. That is, they are time invariant. • In the time series literature, such a stochastic process is known as a weakly stationary. A stationary time series will tend to return to its mean ( called mean reversion) and fluctuate around this mean.

  7. Stationary Stochastic Processes, continued • Properties of stationarity: Let Yt be a stochastic time series Mean: Variance: Covariance:

  8. Stationary Stochastic Processes, continued • A time series is strictly stationary if all the moments of its probability distribution and not just the first two (i.e., mean and variance) are invariant over time. If, however, the stationary process is normal, the weakly stationary stochastic process is also strictly stationary, for the normal stochastic process is fully specified by its two moments, the mean and variance. • Nonstationary time series: if a time series is not stationary in the sense just defined, it is called a nonstationary time series. In other words, a nonstationary time series will have a time-varying mean or a time-varying variance or both.

  9. Stationary Stochastic Processes, continued • White Noise: a special case of stationary stochastic process. • We call a stochastic process purely random or white noise if it has zero mean, constant variance and is serially uncorrelated. IID: identically and independently distributed

  10. 3. Nonstationary stochastic process • Random walk model (RWM): a classical example of nonstationary time series • The term random walk is often compared with a drunkard’s walk. Leaving a bar, the drunkard moves a random distance utat time t, and continuing to walk indefinitely, will eventually drift farther and farther away from the bar. The same is said about stock prices. Today’s stock price is equal to yesterday’s stock price plus a random shock.

  11. 4. Unit Root Process • If, , it becomes a RWM (without drift). If is in fact 1, we face what is known as the unit root problem, that is, a situation of nonstationary; we already know that in this case the variance of Yt is not stationary. The name unit root is due to the fact that . • Thus the terms nonstationarity, random walk, and unit root can be treated as synonymous.

  12. 5. Spurious Regression Again • If Y and X have unit roots then all the usual regression results might be misleading and incorrect. • One way to guard against it is to find out if the time series are cointegrated. • An is a good rule of thumb to suspect that the estimated regression is spurious.

  13. 6.Tests of Nonstationarity • How do we find out whether a given time series is stationary? • At the informal level, weak stationarity can be tested by the correlogram of a given time series. • At the formal level, stationarity can be checked by finding out if the time series contains a unit root.

  14. Tests of Nonstationarity: the correlogram test • Autocorrelation function (ACF): • The correlogram is a graph of autocorrelation at various lags for a given time series • For stationary time series, the corelogram tapers off quickly, whereas for nonstationary time series it dies off gradually. • Q statistic: testing the joint hypothesis that all the up to certain lags are simultaneously equal to zero.

  15. Tests of Nonstationarity: the Unit Root Test • The Unit Root Test • If , then , that is we have a unit root, meaning the time series under consideration is nonstationary.

  16. Tests of Nonstationarity: Dickey-Fuller (DF) Test • The DF test is estimated in three different forms. It was assumed that the error term was uncorrelated.

  17. Dickey-Fuller Test, continued Suppose Yt can be described by the following equation: • Using OLS to run the unrestricted regression: • Using OLS to run the restricted regression: or • Using the sums of squared residuals in the restricted and unrestricted regressions to calculate F statistic, then compare F value with the critical value.

  18. Tests of Nonstationarity: the Augmented Dickey- Fuller (ADF) Test • ADF test is developed if the are correlated. Under the circumstance, the unit root test is run the same way as before.

  19. 7. Cointegration • Regressing one random walk against another can lead to spurious results. • Differencing variables before using them in a regression may result in a loss of long-run information. • Cointegration means that despite two or more time series follow random walks, a linear combination of them can be stationary. If this is the case, we say that these time series are co-integrated. • The AEG,augmented Engle-Granger test,and other tests can be used to find out whether two or more time series are cointegrated. • Cointegration of two or more time series suggests that there is a long-run, or equilibrium, relationship between them.

  20. Testing for Cointegration ( ADF &CRDW Tests) Testing whether there is a co-integrated relationship between two time series. • Step 1: using the ADF test to confirm that variables in the regression are random walks. • Step 2: using OLS to estimate the regression equation: , then tests whether the residuals from the above co-integrating regression are stationary. 1. Perform Augmented Dickey-Fuller unit root test on the residual series to see if the residual is stationary. 2. Look at the Durbin-Watson statistic from the above regression: , compare DW value with the corresponding critical vale at proper confidence level to decide whether reject or accept the null hypothesis.

  21. 8. Error Correction Mechanism • Granger representation theorem: if two variables Y and X are cointergated, then the relationship between the two can be expressed as ECM, the error correction mechanism. where is the error obtained from the regression model with Y and X (i.e. ) and is the error in the ECM model. • The ECM says that depends on - an intuitively sensible point (i.e. changes in X cause Y to change). • In addition, depends on . This latter aspects is unique to the ECM and gives it its name.

  22. Error Correction Mechanism, continued Remember that can be thought of an an equilibrium error. If it is non-zero, then the model is out of equilibrium. Consider the case where and is positive. The latter implies that is too high to be in equilibrium (i.e. is above its equilibrium level of ). Since the term will be negative and so will be negative. In other words, if is above its equilibrium level, then it will start falling in the next period and the equilibrium error will be “corrected” in the model; hence the term “ error correct model”. In the case where the opposite will hold (i.e. is below its equilibrium level, hence which causes to be positive, triggering Y to rise in period t).

  23. Error Correction Mechanism, continued • A distinctive feature of the model is that the ECM has both long run and short run properties built into it. • We use this error term, to tie the short-run behaviour of Y to its long-run value.The ECM developed by Engle and Granger is a means of reconciling the short-run behaviour of an economic variable with its long-run behaviour.

  24. 9. Causality in Economics: the Granger Causality Test The test involves estimating the following pair of regressions:

  25. Granger Causality Test, continued • 1. Unidirectional causality from X to Y is indicated if the estimated coefficients on the lagged X in (1) are statistically different from zero as a group and the set of estimated coefficients on the lagged Y in (2) is not statistically different from zero; • 2. Conversely, unidirectional causality from Y to X exists if the set of lagged X coefficients in (1) is not statistically different from zero and the set of the lagged Y coefficients in (2) is statistically different from zero;

  26. Granger Causality Test, continued 3. Feedback, or bilateral causality, is suggested when the sets of X and Y coefficients are statistically different from zero in both regression; 4. Finally, independence is suggested when the sets of X and Y coefficients are not statistically significant in both the regression.

  27. Granger Causality Test, continued • 1. Regress current Y on all lagged Y terms, but not include the lagged X variables, obtain the restricted residual sum of squares, RSSR. • 2. Run the regression including the lagged X terms, obtain the unrestricted residual sum of squares, RSSu • 3. The null hypothesis is , that is, lagged X terms do not belong in the regression.

  28. Granger Causality Test, continued • 4. To test this hypothesis, we apply the F test Which follows the F distribution with m and (n-k) df. m is equal to the number of lagged X terms and k is the number of parameters estimated in the unrestricted regression. • 5. If the computed F value exceeds the critical F value at the chose level of significance, we reject the null hypothesis, in which case the lagged X terms belong in the regression. This is another way of saying that X Granger-causes Y. • 6. Steps 1 to 5 can be repeated to test model 2, that is, if Y Granger-causes X.

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