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**Regression with Time Series Data**

Judge et al Chapter 15 and 16

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Distributed Lag

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**Polynomial distributed lag**

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**Estimating a polynomial distributed lag**

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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:

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**The Koyck Transformation**

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**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.

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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

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Stationary Processes

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**Non-stationary processes**

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**Non-stationary processes with drift**

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**Summary of time series processes**

Random walk Random walk with drift Deterministic trend

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**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.

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Real data

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Spurious correlation Two random walks we observed earlier.

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**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

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**Checking/testing for stationarity**

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

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**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.

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**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)

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**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

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**Example of a Dickey Fuller Test**

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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.

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**Example of a cointegration test**

Model 1% 5% 10% 3.90 3.34 3.04

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