Applied Econometrics 31 456

1. Applied Econometrics 31 456 This half of the course provides an introduction to stationary time series data. Nobel Prize for Economics in 2003 awarded to Rob Engle and Clive Granger, who highlighted the importance of stationarity in time series data. There are substantial implications for empirical modeling with time series data which is not stationary. Reading: Thomas 13.1 Stationary and non-stationary stochastic processes

2. Properties of Time Series I: Stationary time series Xt is stationary if the series exhibits mean reversion i.e. fluctuates around a constant long run mean. Xthas finite variance which is not dependent upon time. Covariance between two values of Xt depends only on the difference apart in time. E(Xt) = μ(mean is constant in t) Var(Xt) = σ2(variance is constant in t) Cov(Xt ,Xt+k) = χ(k)(covariance is constant in t)

3. Stationary time series WHITE NOISE PROCESS Xt = ut ut ~ IID(0, σ2 )

4. Stationary time seriesXt = 0.5*Xt-1 + utut ~ IID(0, σ2 )

5. Non-stationary time series In contrast a non-stationary time series has the following characteristics (1) Does not have a long run mean which the series returns (2) Variance is dependent upon time and goes to infinity as the sample period approaches infinity (3) Correlogram does not die out - long memory

6. Non-stationary time series UK GDP (Yt) The level of GDP (Y) is not constant and the mean increases over time. Hence the level of GDP is an example of a non-stationary time series.

7. Non-stationary time series RANDOM WALK Xt = Xt-1 + utut ~ IID(0, σ2 ) Mean: E(Xt) = E(Xt-1) (mean is constant in t) X1 = X0+ u1(take initial value X0) X2 = X1 + u2 = (X0 + u1 ) + u2 … Xt = X0 + u1 + u2 +…+ ut E(Xt) = E(X0 + u1 + u2 +…+ ut)(take expectations) = E(X0)= constant

8. Non-stationary time series RANDOM WALK Xt = Xt-1 + utut ~ IID(0, σ2 ) Xt = X0 + u1 + u2 +…+ ut Variance: Var(Xt) = Var(X0) + Var(u1) +…+ Var(ut) = 0 + σ2 +…+ σ2 = t σ2 (variance is not constant through time)

9. Non-stationary time series: Random WalkXt = Xt-1 + utut ~ IID(0, σ2 )

10. Constant covariance - use of correlogram Covariance between two values of Xt depends only on the difference apart in time for stationary series. Cov(Xt ,Xt+k) = χ(k)(covariance is constant in t) (A) Correlation for 1980 and 1985 is the same as for 1990 and 1995. (i.e. t = 1980 and 1990, k = 5) (B) Correlation for 1980 and 1987 is the same as for 1990 and 1997. (i.e. t = 1980 and 1990, k = 7)

11. Non-stationary time series UK GDP (Yt) However, the level of a economic time series is typically non-stationary. The level of GDP (Y) is not constant and the mean increases over time.

12. Non-stationary time series UK GDP (Yt) - correlogram For non-stationary series the Autocorrelation Function (ACF) declines towards zero at a slow rate as k increases.

13. Stationary time series First difference of GDP is stationary ΔYt =Yt -Yt-1 - Growth rate is reasonably constant through time. Variance is also reasonably constant through time

14. Stationary time series UK GDP Growth (Δ Yt) - correlogram Sample autocorrelations decline towards zero as k increases. Decline is rapid for stationary series.

15. Non-stationary Time Series: summary Relationship between stationary and non-stationary process AutoRegressive AR(1) process Xt = α + ρXt-1 + ut ut ~ IID(0, σ2 ) ρ < 1 stationary process - “process forgets past” ρ = 1 non-stationary process - “process does not forget past” α = 0 without drift α  0 with drift

16. Stationary time series with driftXt = α + 0.5*Xt-1 + utut ~ IID(0, σ2 )

17. Non-stationary time series: Random Walk with DriftXt = α + Xt-1 + utut ~ IID(0, σ2 )

18. Time Series Models: summary General Models AutoRegressive AR(1) process without drift Xt = ρXt-1 + ut ρ < 1 stationary process - “process forgets past” ρ = 1 non-stationary process - “process does not forget past” AutoRegressive AR(k) process without drift Xt = ρ1Xt-1 + ρ2Xt-2 + ρ3Xt-3 + ρ4Xt-4 +…+ ρkXt-k + ut