10) Time series econometrics
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1 30C00200 Econometrics 10) Time series econometrics Timo Kuosmanen Professor, Ph.D. 1
2 Topics today Static vs. dynamic time series model Suprious regression Stationary and nonstationary time series Unit root and cointegration Forecasting 2
3 Static vs. Dynamic time series models Static model y t = β 1 + β 2 x t + ε t, t=1,,t. Allow for autocorrelation: Cov[ε t, ε s ] 0 Dynamic models Lagged explanatory variable(s) y t = β 1 + β 2 x t + β 3 x t-1 + ε t, t=1,,t. Lagged dependent variable(s) y t = β 1 + β 2 x t + γy t-1 + ε t, t=1,,t.
4 Autocorrelation or specification error? Mizon (1995) argues that autocorrelation is just an artifact of model misspecification: [Mizon (1995): A note to autocorrelation correctors: Don t J. Econometrics.] The static model with AR(1) disturbances y t = β 1 + β 2 x t + ε t, ε t = ρε t-1 + u t is in fact equivalent to the following dynamic model y t = ρy t-1 + β 1 (1 - ρ) + β 2 (x t - ρx t-1 )+ u t,
5 Equivalence The static model with AR(1) disturbances y t = β 1 + β 2 x t + ε t ε t = ρε t-1 + u t y t = β 1 + β 2 x t + ρε t-1 + u t Note: ε t = y t (β 1 + β 2 x t ), ε t-1 = y t-1 (β 1 + β 2 x t-1 ) Hence, y t = β 1 + β 2 x t + ρ(y t-1 (β 1 + β 2 x t-1 )) + u t Reorganizing the terms gives us the dynamic model y t = ρy t-1 + β 1 (1 - ρ) + β 2 (x t - ρx t-1 )+ u t,
6 Autocorrelation or specification error? The static model with AR(1) disturbances y t = β 1 + β 2 x t + ε t, ε t = ρε t-1 + u t is equivalent to the following dynamic model y t = ρy t-1 + β 1 (1 - ρ) + β 2 (x t - ρx t-1 )+ u t, Instead of correcting for autocorrelation, we could estimate the more general dynamic model y t = β 1 + β 2 x t + β 3 x t-1 + ρy t-1 + ε t, known as the first order autoregressive distributed lag model: ARDL(1,1).
7 Dynamic models Lagged variables y t-1, x t-1 can be included explicitly in the model as explanatory variables Delayed effects Expectations Path dependence
8 Example ARDL(1,1) model Model A: var8 = inflation, var9 = sunspot number. regress var8 var9 var8_01 var9_01 Source SS df MS Number of obs = 151 F( 3, 147) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.0886 var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e var8_ var9_ e-06 _cons
9 Spurious regression Consider the regression model y t = β 1 +β 2 x t + ε t If both y t and x t exhibit stochastic or deterministic trends, estimated β 2 is likely to appear statistically significant, even if y t and x t were completely independent (Granger and Newbold 1974) This apparent relationship (when no actual relationship exist) is referred to as spurious regression (also referred to as spurious correlation) 9
10 A trend can be deterministic, Example, linear time trend y t = μ + βt + ε t Time trends Or stochastic, Example, first-order autoregressive random walk with the drift y t = μ + y t-1 + ε t See the Excel file simulated trends.xls for an illustration 10
11 Stationarity A stochastic process is stationary (or strict(ly) stationary, strong(ly) stationary) if its probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance also do not change over time. Hence, the mean and the variance of the process do not follow trends. Time series that have a deterministic or stochastic trend are nonstationary. Note: Weak stationarity or covariance stationarity only requires that the 1st moment and covariance do not change over time. 11
12 Stationary random process A stationary random process can be stated as y t = μ + ε t where disturbances ε t are referred to as white noise if E(ε t ) = 0 for all t, Var(ε t ) = σ 2 for all t, Cov(ε t, ε s ) = 0 for all t,s Hence E(y t ) = μ, Var(y t ) = σ 2 for all t A stationary random process is said to be integrated of order zero I(0) 12
13 Autoregressive (AR) random process Consider again the first-order autoregressive (AR(1)) process y t = μ + y t-1 + ε t t = 1,,T where disturbances ε t are white noise Substituting the RHS of y t-1 = μ + y t-2 + ε t-1 y t = μ + (μ + y t-2 + ε t-1 ) + ε t = μ + (μ + (μ + y t-3 + ε t-2 )+ ε t-1 ) + ε t = tμ + ε 1 + ε ε t Note: in AR(1) process a disturbance in period t (ε t ) has a permanent effect on all future values of y t+k, k =1,2, 13
14 Autoregressive (AR) random process First-order autoregressive random walk with a drift y t = μ + y t-1 + ε t t = 1,,T is obviously nonstationary. Note: differencing results as a stationary I(0) series y t y t-1 = μ + ε t Therefore, AR(1) process is said to be difference stationary, or integrated of order one I(1). Lesson: A stochastic trend can be de-trended by using the difference (e.g., GDP growth instead of the level of GDP) 14
15 Unit root Introduce the autoregressive coefficient ρ to the AR(1) process y t = μ + ρy t-1 + ε t t = 1,,T Taking the first differece yields y t y t-1 = μ + (ρ 1)y t-1 + ε t If ρ = 1, then the first difference is I(0), and hence y t is I(1). If -1 < ρ < 1, then the process is stationary If ρ = 1, then the process is non-stationary (stochastic trend) The case ρ = 1 is so imortant that it has a name: unit root. 15
16 Unit root econometrics Unit root is the key for drawing a distinction between deterministic versus stochastic trends. If the process has a unit root, then it has a stochastic trend. => First differencing produces a white noise series If the process does not have a unit root, then it can be modeled using a deterministic trend. => Detrending is the preferred approach 16
17 Dickey-Fuller test of unit root H 0 : Process is I(1) (unit root -> stochastic trend) H 1 : Process is I(0) (no unit root -> deterministic trend) Estimate one of the following models by OLS y t y t-1 = γy t-1 + ε t y t y t-1 = μ + γy t-1 + ε t y t y t-1 = μ + βt + γy t-1 + ε t Test statistic: DF = γ /Std.Err(γ) (Augmented DF test) (note: the usual t-stat) In Stata: introduce a time index (t), and declare data to be time series. tsset t 17
18 Example: GDP deflator p t (price index) 18
19 Dickey-Fuller test of GDP deflator (p t ) Model with drift and time trend: p t p t-1 = μ + βt + γp t-1 + ε t Dickey-Fuller test for unit root Number of obs = 151 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = D.var7 Coef. Std. Err. t P> t [95% Conf. Interval] var7 L _trend _cons Conclusion: H 0 is maintained. Unit root, stochastic trend 19
20 Example: Inflation rate π t = (p t p t-1 )/p t-1 20
21 Dickey-Fuller test of inflation rate π t Model with drift and time trend: π t π t-1 = μ + βt + γπ t-1 + ε t Dickey-Fuller test for unit root Number of obs = 151 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = D.var4 Coef. Std. Err. t P> t [95% Conf. Interval] var4 L _trend _cons Conclusion: H 0 is rejected. No unit root 21
22 Cointegration If series y t and x t both have unit roots, then there may be parameters β 1,β 2 such that y t = β 1 + β 2 x t + ε t where the disturbance term ε t follows a stationary I(0) process. In that case, then series y t, x t are said to be cointegrated Examples of cointegrated series Income and consumption Prices of the same commodity in different countries Short and long term interest rates 22
23 Testing for cointegration Engle & Granger (1987) approach 1) Apply Dickey-Fuller test separately to y and x variables, to test if variables are integrated to the same order. (If not, use first differences and test again ) 2) Estimate the following model by OLS y t = β 1 + β 2 x t + ε t 3) Apply Dickey-Fuller test to OLS residuals e t to test if the unit root can be rejected. Note: Use critical values of Engle&Granger. Conclusion: If (H 0 ) unit root is rejected, then y and x are cointegrated. 23
24 Is inflation rate cointegrated with sunspots? Results OLS regression, Model B: var8 = inflation, var9 = sunspot number, var10 = percentage change of the sunspot number of. regress var8 var9 var10 Source SS df MS Number of obs = 152 F( 2, 149) = 3.44 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e var _cons Estimated model: π t = x t z t Save residuals, apply ADF test (with lags 0 4) 24
25 Is inflation rate cointegrated with sunspots?. dfuller var12, trend regress lags(0) Dickey-Fuller test for unit root Number of obs = 151 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = Critical values (Engle & Yoo, 1987, J. Ectr.) 1% 5% 10% D.var12 Coef. Std. Err. t P> t [95% Conf. Interval] var12 L _trend _cons dfuller var12, trend regress lags(4) Augmented Dickey-Fuller test for unit root Number of obs = 147 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value H 0 is rejected (with or without lags) at 5% significance level. Residuals are stationary. Hence, sunspots and inflation are cointegrated accoding to the Engle & Granger test. Z(t) MacKinnon approximate p-value for Z(t) = D.var12 Coef. Std. Err. t P> t [95% Conf. Interval] var12 L LD L2D L3D L4D _trend _cons
26 Economic forecasting For forecasting purposes, simple time series models that describe the behavior of a variable in terms of its own past have proved effective Univariate AR(1) model y t = μ + ρy t-1 + ε t, where ε t is white noise
27 Economic forecasting Example: forecast inflation based on its past values Does subspot number improve the forecast? Divide the observed sample in two parts: Estimation period: Validation period: (the first 100 years) (the last 51 years) (Then forecast the future periods )
28 AR(1) model of inflation Estimate AR(1) model π t = μ + ρπ t-1 + ε t using data of years regress var8_100 var8_01_100 Source SS df MS Number of obs = 100 F( 1, 98) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var8_100 Coef. Std. Err. t P> t [95% Conf. Interval] var8_01_ _cons
29 ARDL(1,1) model of inflation Estimate AR(1) model π t = μ + β 2 x t + β 3 x t-1 + ρπ t-1 + ε t using data of years regress var8_100 var8_01_100 var9_100 var9_01_100 Source SS df MS Number of obs = 100 F( 3, 96) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.1071 var8_100 Coef. Std. Err. t P> t [95% Conf. Interval] var8_01_ var9_ e var9_01_ e-06 _cons
30 One period ahead inflation forecast by AR(1) model ˆ ˆ tt t 1 ˆ 1
31 One period ahead inflation forecast by ARDL(1,1) model ˆ ˆ ˆ ˆ ˆ t t t tt 1 1 2x 3x 1
32 Measuring precision of forecast Root mean squared error (RMSE) of the one period ahead inflation forecast (51 periods) 1 RMSE ˆ tt 1 t t 1962 RMSE of AR(1) model: 2.803% RMSE of ARDL(1,1) model: 2.784% Conclusion: including sun spot infirmation in the model improves the forecast precision, but only marginally. 32
33 Next time Mon 26 Mar Topic: Panel data models 33
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