DIKTAT METODE PENELITIAN ( Ekonometrika - Quantitative Method Time Series) Oleh : Kumba Digdowiseiso FAKULTAS EKONOMI UNIVERSITAS NASIONAL Quantitative Method Analysis: Time Series (1) Kumba Digdowiseiso Quantitative Method Analysis: Time Series (2) Kumba Digdowiseiso Stationarity • OLS estimates for time series models will only be BLUE if, in addition to the usual OLS assumptions, the time series are stationary. • A stochastic time-series is said to be weakly 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 lag between the two time periods. • i.e. if Yt is a stochastic time series, it is weakly stationary if it has the following statistical properties: Mean: E(Yt )= µ Variance: 2 2 var(Yt )= E(Y t − µ) =σ Covariance: γ k = E[(Yt − µ)(Yt+k − µ)] Where γk , the covariance between 2 variables k time periods apart, only depends on k and not t. • A stationary time series exhibits mean reversion • Do economic time series usually exhibit these properties? Recall the examples of time series that were discussed in the last session 250 200 index 150 price 100 50 IMF commodity commodity IMF 0 Data source: IMF IFS 2011 Monthly interest rates in South Africa 14 12 10 8 % 6 4 2 0 Money market interest rate Government bond yield Data source: IMF IFS 2011 Consumer and producer price indices and broad money suuply in South Africa 160 3,E+06 140 2,E+06 120 100 2,E+06 80 (millions) 60 1,E+06 40 SA Rand 5,E+05 20 0 0,E+00 PPI CPI Broad money supply (right axis) Data source: IMF IFS 2011 ut ~ NIID(0,σ ) Gaussian White Noise 2 4 2 1 2 0 wn1 wn3 0 -1 -2 -2 -3 0 20 40 60 80 100 0 20 40 60 80 100 time time White Noise Process – A stochastic stationary time series • The white noise process can be seen as the basic building block for more complex stochastic DGPs (more next session) – The residuals from all time series regressions should be white noise • This is a purely random stochastic process since it has zero mean. • A white noise can be described as a series of random shocks where the succesive observations are uncorrelated with each other. • Typical feature: the series jumps up and down in an unsystematic way. • The value of ut cannot be predicted from ut-1 u ~ NIID(0,σ ) t 2 Random walk: an example of a non-stationary stochastic process • Economic time series tend not to be stationary • The Random Walk model is a classic example of a non-stationary DGP • Pure Random Walk: = + where u is a white noise error term 푌푡 푌푡−1t 푢푡 = 푡 푡 =1 푡 푌 푡 푢 15 5 10 0 5 rw3 rw1 -5 0 -10 -5 0 20 40 60 80 100 0 20 40 60 80 100 time time Random walk with drift 80 60 • = + + 40 where푌푡 훿is the푌푡−1 drift 푢term푡 and ut is a white noise error term 20 훿 • The drift term represents a 0 0 20 40 60 80 100 stochastic trend time • Stock prices, exchange rates and other assets are said to follow a random walk • In fact, testing whether an asset price follows a random walk is a way if testing the efficient market hypothesis empirically – Why is that? Properties of the Random Walk • Let Yt be generated by a random walk process with no drift: Yt = Yt −1 +ut where ut is generated by a white noiseprocess • Note that the RW model can be rewritten such that Yt is expressed as a sum of the past values of the error term, i.e. The sum of a series of random shocks that persist over time. Yt = Y0 + ∑ut where Y0 is the value of Y at t = 0, the start of the process. • Taking expected values we get: E(Yt )= E(Y0 + ∑ut )= Y0 2 • It can be shown that: var(Yt )= tσ where σ2 is the variance of the error term • While the random walk model is nonstationary, it’s first difference is. (Yt −Yt−1 )= ∆Yt = ut Properties of the Random Walk with Drift • Let Yt now be generated by a random walk process with a drift: Yt = δ +Yt−1 + ut where ut is generated by a white noise process • δ is the drift parameter. Yt drifts upwards or downwards depending on the sign on δ. • It can be shown for the random walk with drift that: E(Yt )= Y0 + t.δ var(Y )= tσ 2 where σ2 is the variance of the error term • Taking the first difference of the model gives: Yt −Yt−1 = ∆Yt = δ + ut • If δ > 0, Y will drift upwards and if δ < 0 Y will exhibit a downward drift Difference stationary stochastic processes • The pure random walk is known as a difference stationary process since the first difference of the model is stationary. i.e. It’s mean and variance are time invariant and the covariance depends only in the length of the lag between observations. • Consider the first difference for the random walk with drift: Yt −Yt −1 = ∆Yt = δ + ut where ut is generated by a white noise process • Yt will exhibit a positive or negative trend depending on the sign on δ. • Since we cannot predict the value of the error term or Yt on the basis of past values we say that the random walk with drift model has a stochastic trend. • ΔYt is a difference stationary process, why? Trend Stationary Stochastic Processes • A time series might exhibit a deterministic trend, that is its value is partially determined by the evolution of time. • Consider the following model where Yt: Yt = β1 + β2t +ut • The mean of Yt,, β1+ β2t, is not constant, but it’s variance, σt is. • Subtracting the mean from Yt removes the deterministic trend and gives a stationary process. • Here, Yt is known as a trend stationary process since the procedure of detrending results in a stationary process. Graphical comparison of time series with a stochastic trend and a deterministic trend 300 200 100 0 0 100 200 300 400 500 time stochastic deterministic Orders of integration If: It is said to Notation be integrated of order… a time series, Yt, is stationary 0 Yt ~ I(0) e.g the white noise process the first difference of a series, Yt, 1 Yt ~ I(1) is stationary e.g. the random walk without drift a time series has to be 2 Yt ~ I(2) differenced twice to make it stationary (i.e the first difference of the first difference of Yt is stationary), Yt has to be differenced d times D Yt ~ I(d) before it becomes stationary, Some Properties of integrated series If then 1 Xt ~I(0) and Yt~I(1) Z(t) = (Xt+Yt)~I(1) 2 Xt~I(d) Z(t) = (a+bXt)~I(d), where a and b are constants 3 Xt ~I(d1) and Yt~I(d2) Z(t) = (aXt+bYt)~I(d2), where d1< d2 * 4 Xt~I(d) and Yt~I(d) Z(t) = (aXt+bYt)~I(d ) * d =d unless Xt and Yt are cointegrated in which case d*<d Why is it important to know the order of integration of a series? • To illustrate why it is important to know the order of integration of series consider: Yt = β1 + β2 Xt +ut x y βˆ = ∑ t t 2 2 where xt = Xt − X and yt = Yt −Y ∑ xt • + • Suppose that Yt is I(0), but Xt isI(1). • Since Xt is not stationary, it’s variance will increase overtime in the expression for the OLS estimators with the result that the estimator for β2 will converge to zero as the time span increases. Spurious regression • A second reason why it is important to establish that series are I(0), i.e. stationary, before we estimate on the basis of OLS is the problem of spurious regression. • Spurious, or nonsense, regression refers to the phenomenon where we might concluded that a relationship between two or more variables nonstationary is highly significant when there is no relationship between them at all. Tests for stationarity – Autocorrelation Function (ACF) and Correlogram • We have already come across the first order autocorrelation function in lecture 6. the ACF at lag K is defined as: ( ) γ cov Yt,Y t+k k − ≤ ρ ≤ ρk = = , 1 k 1 var(Yt ) γ 0 • ρk gives the population autocorrelation function. In practice we can only calculate the sample ACF. γˆ = k ρˆ k γˆ0 2 (Y− Y ) ∑(Yt −Y )(Y t+k −Y ) ∑ t where γˆ = and γˆ0 = k n n • The plot of the sample ACF is known as the sample correlogram Correlogram for a white noise 0.20 0.00 -0.20 0 10 20 30 40 Lag Bartlett's formula for MA(q) 95% confidence bands Correlogram for a random walk 1.00 0.50 0.00 -0.50 0 10 20 30 40 Lag Bartlett's formula for MA(q) 95% confidence bands Unit root tests • Stationarity can also sometimes be expressed as the absence of a unit root in a time series. • Consider the random walk model: Yt = ρYt −1 + ut −1 ≤ ρ ≤1 (d1) where ut is a white noise • If ρ is 1, then the model contains a unit root and the series is I(1), the series Yt is not stationary. • If |ρ|≤1, then it can be shown that the time series Yt is stationary in the sense that we have defined it.