Econometrics 2

What is the meaning of cointegration?

two integrated time series of the same order are said to be cointegrated if there is a long-run equilibrium relationship between the two (even if the two are not stationary). Therefore, regressing one of them on the other is not a spurious regression.

Outline the major steps involved in the application of the Box-Jenkins approach to forecasting

1) Identification of ARIMA(p, d, q) 2) Estimation 3) Diagnostic Check

What is a random walk (model)?

A random walk process (model) generates the next value based on the current value plus a white noise (error term). It is non-stationary, integrated of order 1. It is AR(1) with unit root.

A unit root

A unit root stochastic process means a non-stationary stochastic(random) process. It happens when a root of the characteristic polynomial is equal to 1 or inside the unit circle

What is the difference, if any, between tests of unit root and test for cointegration?

A unit root test is used to check the stationarity of a time series. A cointegrated test is performed to check if two time series are cointegrated, although the Engle-Granger cointegraton test is a unit root test of the residuals obtained from regressing one time series on the other.

What are Dickey-Fuller (DF) and augmented DF tests?

Both are tests for the existence of a unit root. The augmented DF test takes into account the possibility of serial autocorrelation of error terms.

What are the differences between Box-Jenkins and VAR approaches to economic forecasting?

Box-Jenkins methodology analyzes a single time series without relying on any economic theory. The VAR approach is a system of simultaneous equations, describing several endogenous economic variables together, however not fully based on economic theory.

What happens if Box-Jenkins techniques are applied to time series that are nonstationary?

Box-Jenkins methodology is based on the stationarity of the time series it analyzes. Therefore, if applied to nonstationary time series, the results are unreliable.

In what sense is VAR atheoretic?

Compared to simultaneous equations models that are fully based on the theory, the VAR approach is not fully based on the theory. All VAR models should have been based on their past lags and the same number of past lags of all other variables in the model (which are all endogenous).

What is meant by a trend-stationary process (TSP) and a difference-stationary process (DSP)?

If a time series has a deterministic trend, the residuals produced from regressing the time series on the time as an explanatory variable are a stationary time series. Then, the time series is called trend-stationary. If a time series become stationary after taking differences, it is difference-stationary.

What is spurious regression?

If one of the two nonstationary time series that are not cointegrated is regressed on the other, although the regression model is meaningless, the value of R^2 is high, the parameters estimated are highly significant, and most likely the Durbin-Watson d test statistic is close to zero. (an indicator of serial autocorrelation of the error terms.)

What is the connection between the cointegrated and spurious regression?

If two nonstationary time series are integrated of the same order, regressing one of them on the other might be meaningless (a spurious regression), unless the two series are cointegrated. In case of cointegrated time series, the regression is not spurious.

What is the connection between Granger causality tests and VAR modelling?

In their functional form, both are the same. Both have a system of equations based on the same number of lags of all endogenous variables. However, the purpose of using each of them is different. Granger causality is used to find if a time series causes another one. VAR modeling is used for forecasting.

How does one decode how many lags to introduce in a concrete application?

Information criteria like Akaike or Schwarz information criteria can be used. Also, cross-validation methods can be useful, too.

What is the major difference between simultaneuos-equation and Box-Jenkins approaches to economic forecasting?

Simultaneous equations models are systems of equations explaining several phenomena according to the underlying theory. Box-Jenkins methodology mostly is not based on theory and it models a single phenomenon. Therefore, Box-Jenkins method is described as atheoretic.

Weak Stationarity

The weak stationary stochastic process has time-invariant (constant over time) mean and variance and the covariance between the values of the two time points can only be a function of the gap between the two points and not of the time itself

What is Engle-Granger (EG) and augmented EG tests?

They are tests for determining if two time series are co-integrated. They are applications of DF and augmented DF to the residuals of regressing one time series on the other time series to check the stationary of the residuals produced. The critical values are revised since the residuals are not observed but predicted.

What is the error correction mechanism (ECM)? What is its relationship with cointegration?

Two cointegrated time series have a long-run equilibrium relationship between each other. However, in short-run there might be disequilibrium between them. An error correction model (mechanism) brings the two cointegrated time series back to their long-run equilibrium.

what is the difference between a deterministic trend and a stochastic trend?

f the trend of a time series is perfectly forecasted for any given time point, the time series has a deterministic trend. If a nonstationary time series does not have a deterministic trend, then its trend is stochastic.

Integrated time series

if a time series is integrated of order d, it means after taking its first d differences, it becomes stationary. With a number of times it is differences less than d, it is still not stationary

If a time series I(3) how many times would you have to difference it to make it stationary?

three times

Is the variance of a random walk stochastic process infinite?

yes, the variance becomes larger and larger (linearly).