# Econometrics Regression Analysis with Time Series Data

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1 Econometrics Regression Analysis with Time Series Data João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

2 Time Series vs. Cross Sectional y t = β 0 + β 1 x t1 + β 2 x t β k x tk + u t Time series data is recorded sequentially in time We no longer have a random sample of individuals: need new assumptions! Instead, we have one realization of a stochastic (i.e. random) process João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

3 Oil prices (\$ per barrel) Time Series João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

4 Time Series João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

5 Time Series Figure: Dow Jones Industrial Average João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

6 Time Series Example Static Model: relates contemporaneous variables y t = β 0 + β 1 z t + u t 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 João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

7 Finite Distributed Lag Models y t = α 0 + δ 0 z t + δ 1 z t δ q z t q + u t We can call δ 0 the impact propensity: it measures the contemporaneous change in y Given a temporary, 1-period change in z, y returns to its original level in period q+1 We can call δ 0 + δ δq the long-run propensity (LRP): it reflects the long-run change in y after a permanent change in z João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

8 Assumptions for Time-Series Models Assumption TS.1 (Linearity in parameters) The stochastic process {(x t1, x t2,..., y t ) : t = 1, 2,..., n} follows the linear model: y t = β 0 + β 1 x t β k x tk + u t u t are called the errors or disturbance Assumption TS.2 (No Perfect Collinearity or Absence of Multicollinearity) In the sample, none of the independent variables is a linear combination of others Assumption TS.3 (Zero conditional mean of the error or Strict exogeneity) E(u t X) = 0, t = 1, 2,..., n This implies the error term in any given period is uncorrelated with the explanatory variables in all time periods (we do not assume random sampling!) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

9 Unbiasedness of OLS Under TS.1 through TS.3, OLS estimators are Unbiased conditional on X, and therefore unconditionally E(u t X) = 0, t = 1, 2,..., n is a very strong assumption, often not verified Suppose: CrimeRate t = β 0 + β 1 PolicePerCapita t + u t in a given city u would need to be uncorrelated with current, past and future values of PolicePerCapita. We can accept u is uncorrelated with current and past values of the regressor. But clearly, an increase in u today is likely to lead politicians to increase PolicePerCapita in the future! TS.3 fails! Suppose: FarmYield t = β 0 + β 1 Labor t + β 2 Rainfall t + u t for a given farm u would need to be uncorrelated with current, past and future values of Labor. But maybe if last year s u was low (some plague) the farmer will not be able to hire as many workers next year. Ok with Rainfall, it will most likely not affect u João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

10 Unbiasedness of OLS (Cont.) We do not worry if u is correlated with past regressors because we can easily solve this problem: just include past regressors, use a distributed lag model But we cannot have u influencing in any way future regressors! (at least to guarantee unbiasedness) Omitted variable bias can be analyzed in the same way as for a cross-section An alternative assumption, closer to the cross-sectional case is: E(u t x t ) = 0. We would say the x s are contemporaneously exogenous. Contemporaneous exogeneity will only be sufficient in large samples João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

11 Variance of OLS Estimators We need to add an assumption of homoskedasticity in order to be able to derive variances Assumption TS.4 (Homoskedasticity) Var(u t x t ) = σ 2 for t = 1, 2,..., n(compare to TS.4) Unlikely to hold in the model: T billrates t = β 0 + β 1 Inflation t + β 2 Deficit t Assumption TS.5 (No Serial Correlation) Corr(u t, u s X) = Corr(u t, u s ) = 0 for t s The independent variables can be correlated across time! João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

12 Variance of OLS Estimators (Cont.) With TS.1 through TS.5, the OLS variances in the time-series case are the same as in the cross-section case: Var(ˆβ X) = σ 2 (X X) 1 Var( ˆβ j ) = σ 2 SST j (1 R 2 j ) The estimator of σ 2 is also the same and remains unbiased OLS remains BLUE With the additional assumption of normal errors, inference is the same Assumption TS.6 (Normality) The errors are independent of X and are independent and identically distributed as Normal(0,σ 2 ) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

13 Time Series with Trends Economic time series often have a trend If two series are trending together, we can t assume that the relation is causal We must always control for unobserved factors that can cause the trends. Otherwise we have a spurious regression problem João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

14 Models for Trends One possible control is a linear trend y t = α 0 + α 1 t + e t, t = 1, 2,... Another possibility is an exponential trend log(y t ) = α 0 + α 1 t + e t, t = 1, 2,... where α1 is approx. the average growth rate of y log(y t ) log(y t 1 ) (y t y t 1 )/y t 1 Or quadratic trend y t = α 0 + α 1 t + α 2 t 2 + e t, t = 1, 2,... João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

15 Adding trends in a regression We should add a trend (usually linear) to the model if either the dependent variable or the independent variables are trending y t = β 0 + β 1 x t1 + β 2 x t β k t + u t If Assumptions TS.1 to TS.3 hold in this model, leaving the trend out would in general lead to biased estimates of the remaining parameters, specially if the other regressors are trending 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 a constant. The residuals form the detrended series Then perform the regression with detrended variables (don t need intercept, it will equal 0). It will give exactly the same estimates as the regression above João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

16 R 2 with trending data Time-series regressions with trends tend to have a very high R 2 Should therefore look at the R 2 from the regression with detrended data This R 2 better reflects how well the x t s explain y t Can also use an adjusted R 2 from the regression with detrended data João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

17 Beer consumption Seasonality Often time-series data exhibits seasonal behavior Seasonality should be corrected by, e.g., regressing each of the seasonal variables on a set of seasonal dummies Can seasonally adjust before running the regression (take the residuals from the previous regression) Should look at R-squared only on adjusted data (as for trends) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

18 Important types of Stochastic Processes 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 Otherwise the process is said to be nonstationary Stationarity implies that the x t s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

19 Covariance Stationary and Weakly Dependent Processes A stochastic process is covariance stationary if E(x t ) is constant, Var(x t ) is constant and for any t, h 1, Cov(x t, x t+h ) depends only on h and not on t A stationary time series is weakly dependent if x t and x t+h are almost independent as h increases If for a covariance stationary process Corr(x t, x t+h ) 0 as h, we say this covariance stationary process is weakly dependent Need weak dependence to use Laws of Large Numbers and Central Limit Theorems João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

20 An MA(1) Process A moving average process of order one [MA(1)] satisfies: y t = e t + α 1 e t 1, t = 1, 2,... where e t is an i.i.d. sequence with mean 0 and variance σe 2 This is a stationary, weakly dependent sequence as y s 1 period apart are correlated, but 2 periods (and further) apart are not João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

21 An AR(1) Process An autoregressive process of order one [AR(1)] satisfies: y t = ρy t 1 + e t, t = 1, 2,... where e t is 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 Corr(y t, y t+h ) = Cov(y t, y t+h ) σ y σ y = ρ h 1 which becomes small as h increases (check the derivation!) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

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