1. ECON 4551
Econometrics II
Memorial University of Newfoundland
Time-series with stationary variables. Dynamic Models, Autocorrelation and Forecasting
Chapter 9
Adapted from Vera Tabakova’s notes

2. Chapter 9: Dynamic Models, Autocorrelation and Forecasting
9.1 Introduction
9.2 Lags in the Error Term: Autocorrelation
9.3 Estimating an AR(1) Error Model
9.4 Testing for Autocorrelation
9.5 An Introduction to Forecasting: Autoregressive Models
9.6 Finite Distributed Lags
9.7 Autoregressive Distributed Lag Models
Principles of Econometrics, 3rd Edition

3. 9.1 Introduction
Figure 9.1
Principles of Econometrics, 3rd Edition

4. The model so far assumes that the observations are not correlated with one another.
This is believable if one has drawn a random sample, but less likely if one has drawn observations sequentially in time
Time series observations, which are drawn at regular intervals, usually embody a structure where time is an important component.
Principles of Econometrics, 3rd Edition

5. If we cannot completely model this structure in the regression function itself, then the remainder spills over into the unobserved component of the statistical model (its error) and this causes the errors be correlated with one another.
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6. Or lagged effects of explanatory variables
In general, there are many situations where inertia affects a time series
Or lagged effects of explanatory variables
Or autocorrelation is introduced by “massaging” of the original data (interpolation, smoothing when going from monthly to quarterly, etc.)
Or nonstationarity
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7. Wrong functional form or missing variables
But, as with heteroskedasticity, we should suspect first that there is a specification issue
Wrong functional form or missing variables
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8. 9.1 Introduction (9.1) (9.2) (9.3) Three ways to view the dynamics:
Principles of Econometrics, 3rd Edition

9. Figure 9.2(a) Time Series of a Stationary Variable
9.1 Introduction
Figure 9.2(a) Time Series of a Stationary Variable
Assume
Stationarity
For now
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10. 9.1 Introduction
Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’
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11. Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’
9.1 Introduction
Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’
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12. 9.2 Lags in the Error Term: Autocorrelation
9.2.1 Area Response Model for Sugar Cane (bangla.gdt)
(9.4)
(9.5)
(9.6)
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13. 9.2.2 First-Order Autoregressive Errors
(9.7)
(9.8)
But assume v well behaved:
(9.9)
(9.10)
Assume also
(the series is stationary
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14. 9.2.2 First-Order Autoregressive Errors
(9.11) OK
(9.12)
OK, constant
(9.13)
Not zero!!!
…but since we stick to stationary series, still constant
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15. 9.2.2 First-Order Autoregressive Errors
(9.14)
Focus on k=1
(9.15)
aka autocorrelation coefficient
(9.16)
Running simple OLS
On sugarcane example
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16. 9.2.2 First-Order Autoregressive Errors
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17. 9.2.2 First-Order Autoregressive Errors
Figure Least Squares Residuals Plotted Against Time
We see “runs”
What if we saw
“bouncing”?
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18. 9.2.2 First-Order Autoregressive Errors
(9.17)
General covariance formula:
Constant variance
Null expectation of x and y if they are errors
(9.18)
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19. 9.3 Estimating an AR(1) Error Model
The existence of AR(1) errors implies:
The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance.
The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading.
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20. 9.3 Estimating an AR(1) Error Model
As with heteroskedastic errors, you can salvage OLS when your data are autocorrelated.
Now you can use an estimator of standard errors that is robust to both heteroskedasticity and autocorrelation proposed by Newey and West.
This estimator is sometimes called HAC, which stands for heteroskedasticity autocorrelated consistent.
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21. 9.3 Estimating an AR(1) Error Model
HAC is not as automatic as the heteroskedasticity consistent (HC) estimator.
With autocorrelation you have to specify how far away in time the autocorrelation is likely to be significant
The autocorrelated errors over the chosen time window are averaged in the computation of the HAC standard errors; you have to specify how many periods over which to average and how much weight to assign each residual in that average.
That weighted average is called a kernel and the number of errors to average is called bandwidth.
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22. 9.3 Estimating an AR(1) Error Model
Usually, you choose a method of averaging (Bartlett kernel or Parzen kernel) and a bandwidth (nw1, nw2 or some integer). Often your software (example GRETL) defaults to the Bartlett kernel and a bandwidth computed based on the sample size, N.
Trade-off: Larger bandwidths reduce bias (good) as well as precision (bad). Smaller bandwidths exclude more relevant autocorrelations (and hence have more bias) but use more observations to increase precision (smaller variance).
Choose a bandwidth large enough to contain the largest autocorrelations. The choice will ultimately depend on the frequency of observation and the length of time it takes for your system to adjust to shocks.
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23. 9.3 Estimating an AR(1) Error Model
Once GRETL recognizes that your data are a time series, then the robust command will automatically apply the HAC estimator of standard errors with the default values of the kernel and Bandwidth (or with your customized choices)
Again, remember that there are several choices about how to run these corrections (so different software packages can yield slightly different results)
In a way, this correction is just an extension of White’s correction for heteroskedasticity
It is based on a formula that has the OLS formula as a special case
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24. 9.3 Estimating an AR(1) Error Model
Sugar cane example The two sets of standard errors, along with the estimated equation are: The 95% confidence intervals for β2 are:
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25. 9.3.2 Nonlinear Least Squares Estimation
However, as in the case of robust estimation under
heteroskedasticity, the HAC correction corrects
or accounts for autocorrelation but does not
exploit it
We can get a more precise estimator if we exploit the
Idea that some observations are different not because
of the effect of the regressor values, but because of the
effect of the adjacent errors
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26. 9.3.2 Nonlinear Least Squares Estimation
(9.19)
(9.20)
(9.21)
(9.22)
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27. 9.3.2 Nonlinear Least Squares Estimation
(9.23)
(9.24)
(9.25)
Now this transformed nonlinear model has errors that are uncorrelated over time
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28. 9.3.2a Generalized Least Squares Estimation
Solving this nonlinear regression is still based on minimizing the sum least squares, but it cannot be done with close formulae now  (because of the nonlinearity in the parameters) so we call it Generalised Least Squares again It can be shown that nonlinear least squares estimation of (9.24) is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure. Details are provided in Appendix 9A.
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29. 9.3.2a Generalized Least Squares Estimation
The nonlinear least squares estimator only requires that the errors be stable (not necessarily stationary).
Other methods commonly used make stronger demands on the data, namely that the errors be covariance stationary.
Furthermore, the nonlinear least squares estimator gives you an unconditional estimate of the autocorrelation parameter and yields a simple t-test of the hypothesis of no serial correlation.
Monte Carlo studies show that it performs well in small samples as well.
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30. 9.3.2a Generalized Least Squares Estimation
But nonlinear least squares requires more computational power than linear estimation, …though this is not much of a constraint these days.
Nonlinear least squares (and other nonlinear estimators) use numerical methods rather than analytical ones to find the minimum of your sum of squared errors objective function. The routines that do this are iterative.
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31. 9.3.3 Estimating a More General Model
(9.26)
(9.27)
We should then test the above restrictions (testing nonlinear restrictions is a possibility)
(9.28)
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32. 9.3.3 Estimating a More General Model
(9.26)
(9.27)
This restricted model could be estimated with OLS as long as v is
well behaved, because there are no nonlinearities
Check that our intuition of having dynamics given by lagged effects of errors
boils down to having lagged effects of dependent and independent
variables!!!
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33. 9.4 Testing for Autocorrelation
9.4.1 Residual Correlogram
(9.29)
(9.30)
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34. 9.4 Testing for Autocorrelation
9.4.1 Residual Correlogram (more practically…)
(9.31)
(9.32)
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35. 9.4.1 Residual Correlogram
Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example
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36. 9.4.1 Residual Correlogram
Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example
In GRETL…
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37. 9.4.1 Residual Correlogram
For this nonlinear model, then, the residuals should be uncorrelated
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38. 9.4.1 Residual Correlogram
Figure 9.5 Correlogram for Nonlinear Least Squares Residuals from Sugar Cane Example with “whitened” error
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39. 9.4.2 A Lagrange Multiplier Test
Another way to determine whether or not your residuals
are autocorrelated is to use an LM (Lagrange multiplier) test.
For autocorrelation, this test is based on an auxiliary
regression where lagged OLS residuals are added
to the original regression equation. If the coefficient on the
lagged residual is significant then you conclude that the
model is autocorrelated.
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40. 9.4.2 A Lagrange Multiplier Test
(9.33)
We can derive the auxiliary regression for the LM test:
(9.34)
First result from GRETL but it comes from the
Rearranged regression format with residual as the dependent variable
Contrary to what the book suggests!!!
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41. 9.4.2 A Lagrange Multiplier Test
(9.35)
Centered around zero, so the power of this test now would come from
The lagged error, which is what we care about…
Second result from GRETL
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42. 9.4.2 A Lagrange Multiplier Test
(9.35)
This was the Breusch-Godfrey LM test for autocorrelation and it is
distributed chi-sq
In STATA:
regress la lp
estat bgodfrey
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43. 9.4.2 A Lagrange Multiplier Test
There are other variants of this test
Note that it is only valid in large samples (theoretically infinitely big ones)
GRETL also reports
Ljung-Box Q‘ test, which is good even if you do not have a regression to work with and just want to analyze any given time series but it is less powerful than Breusch-Godfrey’s test when the null hypothesis of no autocorrelation is false…
The number of lags gives you the df for the chi-sq and the smaller df for the F distributions you need
See appendix for popular (if infamous!) Durbin-Watson test and variants
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44. Stata code * --------------------------------------------------
* LM test
regress la lp
estat bgodfrey
* Note: we set ehat(1) = 0 in order to match result in text. In practice this is unnecessary.
predict ehat, residual
gen ehat_1=L.ehat
replace ehat_1 = 0 in 1
regress ehat lp ehat_1
di (e(N))*e(r2)
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45. GRETL code
You should be able to work it out from these notes and by comparing to the STATA code!!!
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46. 9.5 An Introduction to Forecasting: Autoregressive Models
In general, p should be large enough so that vt is white noise
(9.36)
Adding lagged values of y can serve to eliminate the error autocorrelation
(9.37)
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47. 9.5 An Introduction to Forecasting: Autoregressive Models
Figure 9.6 Correlogram for Least Squares Residuals from AR(3) Model for Inflation
Helps decide
How many lags
To include in the
model
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48. 9.5 An Introduction to Forecasting: Autoregressive Models
(9.38)
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49. 9.5 An Introduction to Forecasting: Autoregressive Models
(9.39)
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50. 9.5 An Introduction to Forecasting: Autoregressive Models
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51. 9.5 An Introduction to Forecasting: Autoregressive Models
(9.40)
(9.41)
(9.42)
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52. 9.5 An Introduction to Forecasting: Autoregressive Models
(9.43)
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53. 9.6 Finite Distributed Lags
This model is just a generalization of the ones
previously discussed.
In this model you include lags of the dependent variable
(autoregressive) and the contemporaneous and lagged
values of independent variables as regressors
(distributed lags).
The acronym is ARDL(p,q) where p is the maximum
distributed lag and q is the maximum autoregressive lag
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54. 9.6 Finite Distributed Lags
(9.44)
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55. 9.6 Finite Distributed Lags
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56. 9.6 Finite Distributed Lags
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57. 9.7 Autoregressive Distributed Lag Models
(9.45)
(9.46)
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58. 9.7 Autoregressive Distributed Lag Models
Figure 9.7 Correlogram for Least Squares Residuals from Finite Distributed Lag Model
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59. 9.7 Autoregressive Distributed Lag Models
(9.47)
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60. 9.7 Autoregressive Distributed Lag Models
Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model
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61. 9.7 Autoregressive Distributed Lag Models
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62. 9.7 Autoregressive Distributed Lag Models
Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model
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63. Keywords autocorrelation interim multiplier
autoregressive distributed lag models
lag length
lagged dependent variable
autoregressive error
LM test
autoregressive model
nonlinear least squares
correlogram
sample autocorrelation function
delay multiplier
standard error of forecast error
distributed lag weight
total multiplier
dynamic models
form of LM test
finite distributed lag
forecast error
forecasting
HAC standard errors
impact multiplier
infinite distributed lag
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64. Chapter 9 Appendices Appendix 9A Generalized Least Squares Estimation
Appendix 9B The Durbin Watson Test
Appendix 9C Deriving ARDL Lag Weights
Appendix 9D Forecasting: Exponential Smoothing
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65. Appendix 9A Generalized Least Squares Estimation
Principles of Econometrics, 3rd Edition

66. Appendix 9A Generalized Least Squares Estimation
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67. Appendix 9A Generalized Least Squares Estimation
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68. Appendix 9A Generalized Least Squares Estimation
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69. Appendix 9B The Durbin-Watson Test
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70. Appendix 9B The Durbin-Watson Test
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71. Appendix 9B The Durbin-Watson Test
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72. Appendix 9B The Durbin-Watson Test
Figure 9A.1:
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73. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
The Durbin-Watson test.
used to be the standard and is exact (it works in small samples)
comes out of almost every software package’s default output
it is easy to calculate
but the critical values depend on the values of the variables=> unless you have a computer to look up the exact distribution…you are stuck
DW did provide the upper and lower distributions but then test may end up being inconclusive
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74. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
Figure 9A.2:
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75. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
if the test is inconclusive.
The size of the inconclusive area shrinks with sample size
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76. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
The Durbin-Watson test.
used to be the standard, but…:
it cannot handle lagged values of the dependent variable (it is biased for autoregressive moving average models, so that autocorrelation is underestimated)
But for large samples one can compute the unbiased normally distributed h-statistic or Durbin’s alternative test statistic )
But then Breusch-Godfrey test is much more powerful instead
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77. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
The Durbin-Watson test.
used to be the standard, but…:
it assumes normality of the errors
it cannot handle more than one lag (it detects only AR1 autocorrelation)
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78. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
Also remember that the Durbin-Watson test assumes
that that there is an intercept in the model
that the X regressors are nonstochastic
that there are no missing values in the series
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79. Appendix 9C Deriving ARDL Lag Weights
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80. Appendix 9C 9C.1 The Geometric Lag
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81. Appendix 9C 9C.1 The Geometric Lag
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82. Appendix 9C 9C.1 The Geometric Lag
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83. Appendix 9C 9C.1 The Geometric Lag
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84. Appendix 9C 9C.2 Lag Weights for More General ARDL Models
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85. Appendix 9D Forecasting: Exponential Smoothing
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86. Appendix 9D Forecasting: Exponential Smoothing
Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α
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