Lecture_23_Prof._Arkoac's_Slides_(Ch_14-15.6)_Fall_10

# Lecture_23_Prof._Arkoac's_Slides_(Ch_14-15.6)_Fall_10 -...

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Time Series Regression II (Fall 2010) Lecture 23 Seyhan Erden Arkonac, PhD Solution to Problem Set 8 is posted Problem Set 9 is due at the beginning of class on Thursday December 9 th . 1

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2 Autoregressions (SW Section 14.3) A natural starting point for a forecasting model is to use past values of Y (that is, Y t –1 , Y t –2 ,…) to forecast Y t . An autoregression is a regression model in which Y t is regressed against its own lagged values. The number of lags used as regressors is called the order of the autoregression. In a first order autoregression , Y t is regressed against Y t –1 In a p th order autoregression , Y t is regressed against Y t –1 , Y t –2 ,…, Y t p .
3 The First Order Autoregressive (AR(1)) Model The population AR(1) model is Y t = 0 + 1 Y t –1 + u t 0 and 1 do not have causal interpretations if 1 = 0, Y t –1 is not useful for forecasting Y t The AR(1) model can be estimated by OLS regression of Y t against Y t –1 Testing 1 = 0 v. 1 0 provides a test of the hypothesis that Y t –1 is not useful for forecasting Y t

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4 Example : AR(1) model of the change in inflation Estimated using data from 1962:I – 2004:IV: t Inf = 0.017 – 0.238 Inf t –1 2 R = 0.05 (0.126) (0.096) Is the lagged change in inflation a useful predictor of the current change in inflation? t = –.238/.096 = –2.47 > 1.96 (in absolute value) Reject H 0 : 1 = 0 at the 5% significance level Yes, the lagged change in inflation is a useful predictor of current change in inflation–but the 2 R is pretty low!
5 Example : AR(1) model of inflation – STATA First, let STATA know you are using time series data generate time=q(1959q1)+_n-1; _n is the observation no. So this command creates a new variable time that has a special quarterly date format format time %tq; Specify the quarterly date format sort time; Sort by time tsset time; Let STATA know that the variable time is the variable you want to indicate th time scale

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6 Example : AR(1) model of inflation – STATA, ctd. . gen lcpi = log(cpi); variable cpi is already in memory . gen inf = 400*(lcpi[_n]-lcpi[_n-1]) ; quarterly rate of inflation at an annual rate T his creates a new variable, inf, the “nth” observation of which is 400 times difference bet ween the nth observation on lcpi and the “n - 1”th observation on that is, the first difference of lcpi compute first 8 sample autocorrelations . corrgram inf if tin(1960q1,2004q4) , noplot lags(8); LAG AC PAC Q Prob>Q ----------------------------------------- 1 0.8359 0.8362 127.89 0.0000 2 0.7575 0.1937 233.5 0.0000 3 0.7598 0.3206 340.34 0.0000 4 0.6699 -0.1881 423.87 0.0000 5 0.5964 -0.0013 490.45 0.0000 6 0.5592 -0.0234 549.32 0.0000 7 0.4889 -0.0480 594.59 0.0000 8 0.3898 -0.1686 623.53 0.0000 if tin(1962q1,2004q4) is STATA time series syntax for using only observations bet 1962q1 and 1999q4 (inclusive). The “tin(.,.)” option requires defining the time first, as we did above
7 Example : AR(1) model of inflation – STATA, ctd . gen dinf = inf[_n]-inf[_n-1]; . reg dinf L.dinf if tin(1962q1,2004q4), r; L.dinf is the first lag of dinf Linear regression Number of obs = 172 F( 1, 170) = 6.08 Prob > F = 0.0146 R-squared = 0.0564 Root MSE = 1.6639 ------------------------------------------------------------------------------ | Robust dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dinf | L1. | -.2380348 .0965034 -2.47 0.015 -.4285342 -.0475354 _cons | .0171013 .1268831 0.13 0.893 -.2333681 .2675707 ------------------------------------------------------------------------------ . dis "Adjusted Rsquared = " _result(8); Adjusted Rsquared = .05082278

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8 Forecasts: terminology and notation Predicted values are “in-sample” (the usual definition) Forecasts
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## This note was uploaded on 11/10/2011 for the course ECON 3142 taught by Professor Arkonac during the Spring '11 term at Columbia.

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Lecture_23_Prof._Arkoac's_Slides_(Ch_14-15.6)_Fall_10 -...

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