# Regression with Time Series Data

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Regression with Time Series Data
Judge et al Chapter 15 and 16

Distributed Lag

Polynomial distributed lag

Estimating a polynomial distributed lag

Geometric Lag Impact Multiplier: change in yt when xt changes by one unit: If change in xt is sustained for another period: Long-run multiplier:

The Koyck Transformation

Autoregressive distributed lag
ARDL(1,1) ARDL(p,q) Represents an infinite distributed lag with weights: Approximates an infinite lag of any shape when p and q are large.

Stationarity The usual properties of the least squares estimator in a regression using time series data depend on the assumption that the variables involved are stationary stochastic processes. A series is stationary if its mean and variance are constant over time, and the covariance between two values depends only on the length of time separating the two values

Stationary Processes

Non-stationary processes

Non-stationary processes with drift

Summary of time series processes
Random walk Random walk with drift Deterministic trend

Trends Stochastic trend Deterministic trend Random walk
Series has a unit root Series is integrated I(1) Can be made stationary only by first differencing Deterministic trend Series can be made stationary either by first differencing or by subtracting a deterministic trend.

Real data

Spurious correlation Two random walks we observed earlier.

Spurious regression R2 0.7495 D-W 0.0305 Variable DF B Value Std Error
T ratio Approx prob Intercept 1 0.5429 26.162 0.0001 RW2 R D-W

Checking/testing for stationarity
Correlogram Shows partial correlation observations at increasing intervals. If stationary these die away. Box-Pierce Ljung-Box Unit root tests

Unit root test If || < 1, then the AR(1) process is stationary.
We can test for nonstationarity by testing the null hypothesis that  = 1 against the alternative that , or simply  < 1.

Dickey Fuller Tests Allow for a number of possible models
Drift Deterministic trend Account for serial correlation Drift Drift against deterministic trend Adjusting for serial correlation (ADF)

Critical values Table 16.4 Critical Values for the Dickey-Fuller Test
Model 1% 5% 10% 2.56 1.94 1.62 3.43 2.86 2.57 3.96 3.41 3.13 Standard critical values 2.33 1.65 1.28

Example of a Dickey Fuller Test

Cointegration In general non-stationary variables should not be used in regression. In general a linear combination of I(1) series, eg: is I(1). If et is I(0) xt and yt are cointegrated and the regression is not spurious et can be interpreted as the error in a long-run equilibrium.

Example of a cointegration test
Model 1% 5% 10% 3.90 3.34 3.04

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