Autocorrelation,

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Autocorrelation, Box Jenkins or ARIMA
Forecasting
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Autocorrelation and the Durbin-Watson Test
An autocorrelation is a correlation of the values
of a variable with values of the same variable
lagged one or more periods back. Consequences of
autocorrelation include inaccurate estimates of
variances and inaccurate predictions.
Lagged Residuals i ?i ?i-1
?i-2 ?i-3 ?i-4 1 1.0
2 0.0 1.0 3 -1.0 0.0 1.0
4 2.0 -1.0 0.0 1.0 5 3.0 2.0 -1.0 0.0 1.0
6 -2.0 3.0 2.0 -1.0 0.0 7 1.0 -2.0 3.0 2.0 -1.0
8 1.5 1.0 -2.0 3.0 2.0 9 1.0 1.5 1.0 -2.0 3.0
10 -2.5 1.0 1.5 1.0 -2.0
The Durbin-Watson test (first-order
autocorrelation) H0 ?1 0
H1??? ? 0 The Durbin-Watson
test statistic
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DW d Test
4 Steps
Step 1 Estimate
And obtain the residuals
Step 2 Compute the DW d test statistic
Step 3 Obtain dL and dU the lower and upper
points from the Durbin-Watson tables
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Step 4 Implement the following decision rule
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Critical Points of the Durbin-Watson Statistic
?0.05, n Sample Size, k Number
of Independent Variables
k 1 k 2 k 3 k 4 k 5
n dL dU dL dU dL dU dL dU dL dU
15 1.08 1.36 0.95 1.54 0.82 1.75 0.69 1.97 0.56 2.
21 16 1.10 1.37 0.98 1.54 0.86 1.73 0.74 1.93 0.6
2 2.15 17 1.13 1.38 1.02 1.54 0.90 1.71 0.78 1.9
0 0.67 2.10 18 1.16 1.39 1.05 1.53 0.93 1.69 0.82
1.87 0.71 2.06 . . . . . .
. . . . . . . . . . . . 65 1.57 1.63
1.54 1.66 1.50 1.70 1.47 1.73 1.44 1.77
70 1.58 1.64 1.55 1.67 1.52 1.70 1.49 1.74 1.46 1
.77 75 1.60 1.65 1.57 1.68 1.54 1.71 1.51 1.74 1
.49 1.77 80 1.61 1.66 1.59 1.69 1.56 1.72 1.53 1
.74 1.51 1.77 85 1.62 1.67 1.60 1.70 1.57 1.72
1.55 1.75 1.52 1.77 90 1.63 1.68
1.61 1.70 1.59 1.73 1.57 1.75 1.54 1.78
95 1.64 1.69 1.62 1.71 1.60 1.73 1.58 1.75 1.56 1
.78 100 1.65 1.69 1.63 1.72 1.61 1.74 1.59 1.76 1
.57 1.78
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Durbin-Watson Test for Autocorrelation An Example

• The Banner Rock Company manufactures and markets
  its own rocking chair. The company developed
  special rocker for senior citizens which it
  advertises extensively on TV. Banners market
  for the special chair is the Carolinas, Florida
  and Arizona, areas where there are many senior
  citizens and retired people The president of
  Banner Rocker is studying the association between
  his advertising expense (X) and the number of
  rockers sold over the last 20 months (Y). He
  collected the following data. He would like to
  use the model to forecast sales, based on the
  amount spent on advertising, but is concerned
  that because he gathered these data over
  consecutive months that there might be problems
  of autocorrelation.

Month Sales (000) Ad (millions)
1 153 5.5
2 156 5.5
3 153 5.3
4 147 5.5
5 159 5.4
6 160 5.3
7 147 5.5
8 147 5.7
9 152 5.9
10 160 6.2
11 169 6.3
12 176 5.9
13 176 6.1
14 179 6.2
15 184 6.2
16 181 6.5
17 192 6.7
18 205 6.9
19 215 6.5
20 209 6.4
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Durbin-Watson Test for Autocorrelation An Example

• Step 1 Generate the regression equation

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Durbin-Watson Test for Autocorrelation An Example

• The resulting equation is Y - 43.802 35.95X
• The coefficient (r) is 0.828
• The coefficient of determination (r2) is 68.5
• There is a strong, positive association between
  sales and advertising
• Is there potential problem with autocorrelation?

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Durbin-Watson Test for Autocorrelation An Example
-43.80235.95C3
(E4-F4)2
E42
B3-D3
E3
?(ei)2
?(ei -ei-1)2
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Durbin-Watson Test for Autocorrelation An Example

• Hypothesis Test
• H0 No residual correlation (? 0)
• H1 Positive residual correlation (? gt 0)
• Critical values for d given a0.5, n20, k1
• dl1.20 du1.41

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Autoregressive Models
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Box Jenkinsor Arima Forecasting

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• All stationary time series can be modeled as AR
  or MA or ARMA models
• A stationary time series is one with constant
  mean ( ) and constant variance.
• Stationary time series are often called mean
  reverting seriesthat in the long run the mean
  does not change (cycles will always die out).
• If a time series is not stationary it is often
  possible to make it stationary by using fairly
  simple transformations

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Nonstationary Time series

• Linear trend
• Nonlinear trend
• Multiplicative seasonality

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How to make them stationary

• Linear trend
• Take non-seasonal difference. What is left over
  will be stationary AR, MA or ARMA
• Nonlinear trend
• Exponential growth
• Take logs this makes the trend linear
• Take non--seasonal difference
• Non exponential growth ?
• Take logs
• Multiplicative seasonality often occurs when
  growth is exponential.

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Identification

• What does it take to make the time series
  stationary?
• Is the stationary model AR, MA, ARMA
• If AR(p) how big is p?
• If MA(q) how big is q?
• If ARMA(p,q) what are p and q?

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ARMA models

• If you cant easily tell if the model is an AR or
  a MA, assume it is an ARMA model.

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Box-Jenkins Method

• First of all, the analyst identifies a tentative
  model considering the nature of the past data.
  This tentative model and the data are entered in
  the computer. The Box-Jenkins program then gives
  the values of the parameters included in the
  model. A diagnostic check is then conducted to
  find out whether the model gives an adequate
  description of the data. If the model satisfies
  the analyst in this respect, then it is used to
  make the forecast.