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1 1 1

1

....... ....(1 )

j

t t t t jY e e e

= + + + + +

(2)

Taking expectations of both the sides in equation 2 yields the proposition.

For simplicity, for the remainder of this note, let us set 0= .Proposition 1 then implies

( ) 0t

E Y = .

Proposition 2:

Variance of

22

2

1(1 )

e

t YY

= =

2 2 2 2 2

1 1 1

2 2 2

1

2

2

2

1

var ( ) ( ) ( ) ( ) 2 ( )

(1 )

t t Y t t t t

Y e

e

Y

iance Y E Y E Y E e E Y e

= = = + +

= +

=

Define 1 as 1( )t t E YY

Proposition 3:

2 2

1 1/ (1 )e =

Proof:

2

1 1 2 1

2

1

2 2

1 1

( ) ( ) ( )

/ (1 )

t t t t t

Y

e

E Y Y E Y E Y e

= +

=

=

Proposition 4:


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2 2

1 1( ) / (1 )j j

t t j eE Y Y = =

Exercise: Prove Proposition 4.

Example: Suppose2

1 0.7 and 1e = = . The following graph plotsj as a function of

j=0,1,2,3,,,,,,,,,.

As you can see, the graph trends to zero as j becomes larger. This is referred to as the

autocorrelation function.

Exercise: Plot the autocorrelation function for an AR(1) process for 1 0.7 = and2 2e

= .

Proposition 6:

Define the correlation between tY and t jY asj .

1

j j =

Prove proposition 6.

A plot ofj

against j is referred to as the plot of the autocorrelation function.


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Estimating the parameters of an AR(1) process:

Suppose we want to estimate the parameters 1and in the following AR(1) process:

1 1t t tY Y e = + + where we have available to us T observations on Y.

Then,

1__^ ^

1 1

2 1

( / 1)T T

t t

t t

Y T Y

= =

=

__ __

1 1

^ 2

1 2___

1 1

2

T

t t t t

t

T

t t

t

Y Y Y Y

Y Y

=

=

=

As you can see, this is just like estimating the parameters of a two variable regression, with

1replacing

t tY X

. The standard errors of the coefficients can also be calculated in the same

way that you have done for two variable regression.

Forecasting from an AR(1) process:

Suppose we have the following AR(1) process:

1 1t t tY Y e = + +

The unconditional forecast for tY is simply its unconditional mean,1

(1 )

. The forecast

error for a one period ahead forecast is:

1 1 1 1 1 1 1

1 1 1

( )(1 ) (1 ) (1 )

t t t t t t Y Y e Y e e

+ + +

= + + = + + + +


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The error of two period ahead conditional forecast given tY is given by 1 1t te e+ + and the

variance of the forecast error is given by2 2

1(1 )

e + , i.e, the two period ahead conditional

forecast is less accurate than the one period ahead conditional forecast.

95% confidence intervals for a one period ahead conditional forecast when te is normally

distributed:

forecast +(-)1.96^

e

95% confidence intervals for a two period ahead conditional forecast when te is normally

distributed:

forecast +(-)1.96^ 2

1(1 )

e +

The confidence interval for two period ahead forecast error is wider than that for one period

ahead forecast error.

Exercise: Show that as j tends to infinity, the j period ahead conditional forecast given tY

converges to the unconditional forecast.

Exercise: Show that as j tends to infinity, the variance of the j period ahead forecast

converges to the variance of the error of the unconditional forecast.

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