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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 If change in x t is sustained for another period: Impact Multiplier: change in y t when x t changes by one unit: Long-run multiplier:

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The Koyck Transformation

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Autoregressive distributed lag ARDL(1,1) Represents an infinite distributed lag with weights: ARDL(p,q) 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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AR(1) Random walk Random walk with drift Deterministic trend Summary of time series processes

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Trends Stochastic 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

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Spurious regression VariableDFB ValueStd ErrorT ratioApprox prob Intercept RW 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

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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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Table 16.4 Critical Values for the Dickey-Fuller Test Model1%5%10% Standard critical values Critical values

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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 e t is I(0) x t and y t are cointegrated and the regression is not spurious e t can be interpreted as the error in a long- run equilibrium.

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Example of a cointegration test Model1%5%10%

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