9) Time series econometrics

Transcription

1 30C00200 Econometrics 9) Time series econometrics Timo Kuosmanen Professor Management Science 1

2 Macroeconomic data: GDP Inflation rate Examples of time series Financial market data: Asset returns Stock prices Management Science: Big data (new types of data from internet, social media, etc.): Google Trends Facebook likes 2

3 30E00800 Time Series Analysis (6 cr) Teaching Period: IV-V Spring ( ) Töölö Campus Topics: regression analysis of time series, ARMA models, forecasting, cointegration, ARCH and GARCH models, panel data. Lectures 40 h, Tomi Seppälä Grading: final exam (50%), exercises (20%) and project work (30%). Literature: Brooks, Chris: Introductory econometrics for finance, 2nd edition,

4 How time series and cross sections differ? Cross section i =1,,n represents a randomly drawn sample from the population Time series t = 1,,T describes a (random) path of a variable in a time window [1, T] Time series have a natural chronological order, whereas in a cross section the ordering of observations does not matter Path dependence: correlations across time periods (t, t+1) Autocorrelation (serial correlation) 4

5 Suprious regression: Topics today How time series can mislead you? Jevons (1875) sunspot theory of business cycle revisited Stationary and nonstationary time series Unit root Simulated unit root process Dickey-Fuller test Cointegration Engle and Granger test Sunspot theory revisited again using Dickey-Fuller and Engle-Granger tests 5

6 Jevons sunspot theory of business cycle Motivation: What causes business cycle (booms, recessions)? Economic theory predicts a stable equilibrium. Rationale: Sunspots have cyclical fluctuations -> affect the weather -> affect agricultural production -> affect prices -> economy Empirical evidence? Jevons was among the first economists to try find empirical support for his theory. Econometrics of 1870s. 6

7 Sunspot cycle (Data source: NASA) 7

8 Jevons sunspot theory revisited Dependent variable: Inflation rate in Finland (π t ) Calculated as the annual percentage change of the GDP deflator Statistics Finland (Historical national accounts) Period: Independent variables: International sunspot number (x t ) Annual percentage change of the sunspot number (z t ) Source: NASA, Monthly data since year 1749 Models: A) π t = α + βx t + ε t, t = 1861,,2012 B) π t = α + βx t + γz t + ε t, t = 1861,,2012 8

9 Jevons sunspot theory revisited Results OLS regression Model A: var8 = inflation, var9 = sunspot number. regress var8 var9 Source SS df MS Number of obs = 152 F( 1, 150) = 5.86 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.1261 var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e _cons Estimated model: π t = x t 9

10 Jevons sunspot theory revisited 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 10

11 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 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) 11

12 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 12

13 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. 13

14 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) 14

15 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, 15

16 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) 16

17 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. 17

18 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 18

19 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 = ρy t-1 + ε t y t = μ + ρy t-1 + ε t y t = μ + βt + ρy t-1 + ε t (Augmented DF test) Test statistic: DF = (ρ 1)/Std.Err(ρ) In Stata: introduce a time index (t), and declare data to be time series. tsset t 19

20 Example: GDP deflator p t (price index) 20

21 Dickey-Fuller test of GDP deflator (p t ) Model with drift and time trend: p t = 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 21

22 Example: Inflation rate π t = (p t p t-1 )/p t-1 22

23 Dickey-Fuller test of inflation rate π t Model with drift and time trend: π t = 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 23

24 Cointegration If series y t and x t β 1,β 2 such that both have unit roots, then there may be parameters 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 24

25 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. 25

26 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) 26

27 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 gets 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

28 Next time Wed 7 Oct Time series continued Autocorrelation Dynamic models and lagged variables Forecasting 28