7 Introduction to Time Series

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1 Econ Econometric Review 1 7 Introduction to Time Series 7.1 Time Series vs. Cross-Sectional Data Time series data has a temporal ordering, unlike cross-section data, we will need to changes some of our assumptions to take into account that we no longer have a random sample of individuals Instead, we have one realization of a stochastic (i.e. random) process Y t, t = 1, 2,... For the standard econometric techniques to have the usual properties, time series data need to be stationary, i.e. the means, variances and

2 Econ Econometric Review 2 covariances of the time series data cannot depend on the specific time period Time series data thus raises new technical issues and properties. sort t. tsset t Time lags: the first lag of Y t is Y t 1, the j-th lag is Y t j Correlation over time (serial correlation or autocorrelation): the j-th autocorrelation is Corr(Y t, Y t j ) Correlogram: for a stationary variable, the autocorrelations should die out fairly quickly as j becomes large

3 Econ Econometric Review 3 time variable: t, 1 to 41. corrgram hrwage, noplot lag(8) LAG AC PAC Q Prob>Q Forecasting models that have no causal interpretation (specialized tools for forecasting): autoregressive (AR) models, autoregressive distributed lag (ADL) models

4 Econ Econometric Review 4 Conditions under which dynamic effects can be estimated, and how to estimate them Calculation of standard errors when the errors are serially correlated We are more interested in models where a potentially causal link between variables is sought than in pure forecasting models A static model relates contemporaneous variables: Y t = β 0 + β 1 Z t + u t (1) For example, suppose we want to study the relationship between (log)wages and (log)productivity using yearly data from 1947 through 1987 (see Wooldridge, ex.11.7)

5 Econ Econometric Review Year Log(wages) Log(output per hr) Figure 1: Log(Wages) and Log(Output per hr) We could estimate the relationship by OLS, but we will be suspicious of the very large R-squared

6 Econ Econometric Review 6. reg lhrwage loutphr Source SS df MS Number of obs = F( 1, 39) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhrwage Coef. Std. Err. t P> t [95% Conf. Interval] loutphr _cons

7 Econ Econometric Review 7 Similarly we will be suspicious about the significance of the coefficient of the log(price) of frozen orange juice on log(wages) from 1948 to 1987 and think of it as a spurious relationship. reg lhrwage lfrozen t Source SS df MS Number of obs = F( 2, 37) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhrwage Coef. Std. Err. t P> t [95% Conf. Interval] lfrozen t _cons

8 Econ Econometric Review 8 A finite distributed lag (FDL) model allows one or more variables to affect Y with a lag: Y t = α 0 + δ 0 Z t + δ 1 Z t 1 + δ 2 Z t 2 + u t More generally, a finite distributed lag model of order q will include q lags of Z We can call δ 0 the impact propensity it reflects the immediate change in Y. For a temporary, 1-period change, Y returns to its original level in period q + 1 We can call δ 0 +δ 1 + +δ q the long-run propensity (LRP) it reflects the long-run change in Y after a permanent change

9 Econ Econometric Review 9 It is still appropriate to use OLS when the following assumptions for unbiasedness are satisfied: TS1: the model that is linear in parameters: Y t = β 0 + β 1 X 1t + + β k X kt + u t TS2: the zero conditional mean assumption holds for all X: E(u t X) = 0, t = 1, 2,... n That is, the error term in any given period is uncorrelated with the explanatory variables in all time periods [E(u t X s ) = 0, s = t]. This zero conditional mean assumption implies that all the Xs are strictly exogenous

10 Econ Econometric Review 10 In large sample, contemporaneous exogeneity [E(u t X t ) = 0] can be sufficient. TS3: We still need to assume that no X is constant, and that there is no perfect collinearity. As in the cross-section case, we need to add an assumption of homoskedasticity in order to be able to derive variances, V ar(u t X) = V ar(u t ) = σ 2 We also need the assumption of no serial correlation: Corr(u t, u s X) = 0 for t s.

11 Econ Econometric Review Detrending Time Series Economic time series often have a trend. A deterministic trend is a non-random function of time, while a stochastic trend is random and varies over time. So that two series could be trending together, but with is no causal relation between them. Rather they could be trending because of other unobserved factors. Even if those factors are unobserved, we can control for them by directly controlling for a deterministic trend

12 Econ Econometric Review 12 One possibility is to include a linear trend, which can be modeled as Y t = α 0 + α 1 t + ɛ t, t = 1, 2,... Adding a linear trend term to a regression is the same thing as using detrended series in a regression Detrending a series involves regressing each variable in the model on t and using the residuals as the new detrended series Other functional forms for the trend can also be used. For example, a quadratic trend, which can be modeled as Y t = α 0 +α 1 t+α 2 t 2 +ɛ t, t = 1, 2,...

13 Econ Econometric Review 13. reg lhrwage loutphr t t2 Source SS df MS Number of obs = F( 3, 37) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhrwage Coef. Std. Err. t P> t [95% Conf. Interval] loutphr t t _cons quietly reg lhrwage t t2. predict reswt2, res

14 Econ Econometric Review 14. quietly reg loutphr t t2. predict resot2, res. reg reswt2 resot2 Source SS df MS Number of obs = F( 1, 39) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = reswt2 Coef. Std. Err. t P> t [95% Conf. Interval] resot _cons 1.85e

15 Econ Econometric Review 15 Often time-series data exhibits some periodicity, referred to seasonality For example, quarterly data on retail sales will tend to jump up in the 4th quarter, reflecting Christmas shopping Seasonality can be dealt with by adding a set of seasonal dummies or as with trends, the series can be seasonally adjusted before running the regression

16 Econ Econometric Review Types of Stochastic Processes Stationarity A problem of spurious correlation can arise when OLS is runned on nonstationary data giving a high R-squared and a statistically significant t ratio, even when there is no relationship in the data. The use of OLS to test some hypotheses supposes that the data is stationary. If a series of data is not stationary then it must have some property which tends to drift upwards or downwards as the series unfolds over time.

17 Econ Econometric Review 17 A stochastic process is stationary if for every collection of time indices 1 < t 1 t m the joint distribution of (X t1,..., X tm ) is the same as that of (X t1+h,..., X tm+h ) for h 1 Stationarity says that the past is like the present and the future, at least in a probabilistic sense. For example, a series will be said to be mean stationary, if it fluctuates around its mean level, over time without showing any tendency to drift away from it. A trending series cannot be stationary, since the mean is changing over time

18 Econ Econometric Review 18 A series that is variance stationary fluctuates around its variance, over time, without showing any tendency to drift away form this given level A series that is covariance stationary will fluctuate around a given covariance, over time, without showing any tendency to drift away fro this given level, that is, if E(X t ) is constant, V ar(x t ) is constant and for any t, h 1, Cov(X t, X t+h ) depends only on h and not on t. This weaker form of stationarity requires only that the mean and variance be constant across time, and the covariance just depends on the distance across time

19 Econ Econometric Review Weak Dependence A stationary time series is weakly dependent if X t and X t+h are almost independent as h increases, i.e. in the correlation between X t and X t+h goes to zero sufficiently quickly as h. A moving average process of order one [MA(1)] can be characterized as one where X t = e t + α 1 e t 1, t = 1, 2,... with e t being an i.i.d. sequence with mean 0 and variance σ 2 e. This is a stationary, weakly dependent sequence as variables 1 period apart are correlated, but 2 periods apart they are not

20 Econ Econometric Review 20 An autoregressive process of order one [AR(1)] can be characterized as one where Y t = ρy t 1 + e t, t = 1, 2,... with e t being an i.i.d. sequence with mean 0 and variance σ 2 e. For this process to be weakly dependent, it must be the case that ρ < 1, since Corr(Y t, Y t+h ) = Cov(Y t, Y t+h )/(σ y σ y ) = ρ h which becomes small as h increases A trending series can be weakly dependent. If a series is weakly dependent and is stationary about its trend, we will call it a trend-stationary process We refer to a weakly dependent process as being integrated of order zero, [I(0)], i.e. does not need to be integrated by contrast with series that are differenced.

21 Econ Econometric Review 21 Practically, this means that nothing needs to be done to such series before using them in regression analysis. For asymptotic unbiasedness (consistency), we can weaken the exogeneity assumptions somewhat relative to those for unbiasedness (See Wooldridge, Chap. 11) A process can combine all features and be autoregressive integrated moving average (ARIMA), for example an ARIMA(1,1,1) process would be (1 ρ 1 L) Y t = (1 α 1 L)e t where L is the lag operator, and is the differencing operator. The order of the process depends on the order of lag operator on the series (AR part), of the difference operator on Y t (I part) and of the lag operator on the error part (MA part).

22 Econ Econometric Review Random Walks A random walk is an AR(1) model where ρ 1 = 1, meaning the series is not weakly dependent. A random walk is a special case of what is called a unit root process A random walk is integrated of order one, [I(1)], meaning a first difference will be I(0) The first-differenced series Y t = Y t Y t 1 = e t is actually an i.i.d. sequence

23 Econ Econometric Review 23 With a random walk, the expected value of Y t is always Y 0, it does not depend on t and V ar(y t ) = σ 2 t, so it increases with t We say a random walk is highly persistent (or strongly dependent) since E(Y t+h Y t ) = Y t for all h > 1 In order to use a highly persistent series and get meaningful estimates and make correct inferences, we want to transform it into a weakly dependent process 7.4 Testing for Unit Root Consider an AR(1): Y t = α + ρy t 1 + ɛ t,

24 Econ Econometric Review 24 Assume there is a unit root, let H 0 : ρ = 1, We could define θ = ρ 1, and subtracting Y t 1 from both sides to obtain Y t = α + θy t 1 + ɛ t Unfortunately, a simple t-test for H 0 : θ = 0 vs H 0 : θ < 0 is inappropriate, since this is an I(1) process A Dickey-Fuller Test uses the t-statistic, but different critical values If a series is clearly trending, then we need to adjust for that or we might mistake a trend stationary series for one with a unit root, we can simply add a trend to the model

25 Econ Econometric Review 25. dfuller lhrwage, trend Dickey-Fuller test for unit root Number of obs = Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = dfuller loutphr, trend Dickey-Fuller test for unit root Number of obs = Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) =

26 Econ Econometric Review 26 We cannot reject that log(wages) and log(output per hr) exhibit a unit root Year lhrwage lhrwage[_n 1] loutphr loutphr[_n 1] Figure 2: Growth in Log(Wages) and Log(Output per hr) Given that both the log(wages) and log(output) are integrated series

27 Econ Econometric Review 27 with clear linear trends, it is better to estimate the relationship is differences Indeed, we can reject that the first differences of log(wages) as a unit root, the same for log(output per hr). dfuller D.lhrwage, trend Dickey-Fuller test for unit root Number of obs = Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) =

28 Econ Econometric Review 28. dfuller D.loutphr, trend Dickey-Fuller test for unit root Number of obs = Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = reg D.lhrwage D.loutphr t Source SS df MS Number of obs = F( 2, 37) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.01511

29 Econ Econometric Review 29 D.lhrwage Coef. Std. Err. t P> t [95% Conf. Interval] loutphr D t _cons Cointegration If for two I(1) processes, Y t and X t, there is a β such that Y t βx t is an I(0) process, then we say that that Y and X are cointegrated, and call β the cointegration parameter If we know β, testing for cointegration is straightforward

30 Econ Econometric Review 30 If we define S t = Y t βx t, do Dickey-Fuller test and if we reject a unit root, then they are cointegrated If β is unknown, then we first have to estimate β, which adds a complication After estimating β, we run a regression of û t on û t 1 and compare t-statistic on û t 1 with the special critical values If there are trends, we need to add it to the initial regression that estimates β and use different critical values for t-statistic on û t 1

31 Econ Econometric Review 31. reg lhrwage loutphr t Source SS df MS Number of obs = F( 2, 38) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhrwage Coef. Std. Err. t P> t [95% Conf. Interval] loutphr t _cons predict reswt, res. dfuller reswt, trend

32 Econ Econometric Review 32 Dickey-Fuller test for unit root Number of obs = Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = If û t is stationary, then the unit roots in Y t adn X t have cancelled out, and Y and X are said to be cointegrated Then we can estimate the original model or we may want to specify a more general dynamic model, called a vector autoregressive (VAR) model, where the MA part of ARIMA models is omitted for simplicity (beyond the scope of the review)

33 Econ Econometric Review 33 Here even though we are so far from the 10% critical value, there is evidence that the unit roots in log(wages) and log(output per hr) have not cancelled out completely, the regression of log(wages) and log(output per hr) is likely spurious. We are better off with the model in differences with a trend. reg ghrwage goutphr t Source SS df MS Number of obs = F( 2, 37) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = ghrwage Coef. Std. Err. t P> t [95% Conf. Interval] goutphr

34 Econ Econometric Review 34 t _cons

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