6.4 Prediction-We have already seen how to make predictions about our dependent variable using our OLS estimates and values for our independent variables

-However, these estimates are subject to sampling variation

-a confidence interval is therefore often preferable to a point estimate

6.4 Prediction-Assume we want to obtain the value of y (denoted by θ) given certain independent values xj=cj:

),...,,|(

(6.27) ...

2211

22110

kk

kk

cxcxcxyE

ccc

-since we don’t have actual values for BJ,

our estimator of θ becomes:(6.29) ˆ...ˆˆˆˆ

22110 kkccc

6.4 Prediction-However, due to uncertainty, we may prefer a confidence interval estimate for θ:

)ˆ(*ˆ setCI -Since we don’t know the above standard

error, similar to F tests we construct a new regression using the calculation:

kkccc ...22110

6.4 Prediction-substituting this calculation into the population regression function gives us:

(6.30)u )(

...)()( 222111

kkk cx

cxcxy

-In this regression, the intercept gives us the needed predicted value and intercept to construct our confidence interval-note that all our slope coefficients and their standard errors must be the same as the original regression

6.4 Prediction Notes1) Note that the variance of this

prediction will be smallest at the mean values of xj

-This is due to the fact that we have most confidence in our OLS estimates in the middle of our data.

2) Note also that this confidence interval applies to the AVERAGE observation with the given x values, and does not apply to any PARTICULAR observation

6.4 Individual Prediction-When obtaining a confidence interval for an INDIVIDUAL (often one not in the sample), it is referred to as a PREDICITON INVERVAL and the actual outcome is:

(6.33) u... 00022

0110

0 kk xxxy

-x0 is the individual’s x values (which could be observed) and u0 is the unobserved error-Our best estimate of y0, from OLS, is:

0022

0110

0 ˆ...ˆˆˆˆ kk xxxy

6.4 Individual Prediction-given our OLS estimation, our PREDICITON ERROR is:

ˆ -u)...( ˆ

(6.34) ˆˆ0000

220110

0

000

yxxxe

yye

kk

-We know that E(Bjhat)xj=Bjxj (since OLS is unbiased) and E(u0)=0, therefore

0)ˆ( 0 eE

6.4 Individual Prediction-Note that u0 is uncorrelated with Bjhat, since error comes from the population and OLS estimates are from a sample (and the actual error is uncorrelated with the estimated error)-Furthermore, u0 is uncorrelated with yhat0

-Therefore, the VARIANCE OF THE PREDICIOTN ERROR simplifies to:

200

000

)ˆ()ˆ(

(6.35) )()ˆ()ˆ(

yVareVar

uVaryVareVar

6.4 Individual Prediction-The above variation comes from two sources:1) Variation in yhat0. Since yhat0 comes from estimates of the Bj’s, which have a

variance proportional to 1/n, Var(yhat0) is also proportional to 1/n, and is very small for large samples

2) σ2. Since (1) is often small, the variance of the error is often the dominant term

6.4 Individual Prediction-From our CLM assumptions, Bjhat and u0 are normally distributed, so ehat0 is also normally distributed.-Furthermore, we can rearrange our variance formulas to see that:

(6.36) }ˆ)]{[se(y )ese( 2200

-due to the σhat2 term, individual prediction CI’s are wider than average prediction CI’s and calculated as:

)ˆ(*ˆ 00 esetyCI

6.4 Residual Analysis-RESIDUAL ANALYSIS involves examining individual observations to determine whether

the predicted value is above or below the true value-a negative residual can indicate an observation is undervalued or has an unmeasured

characteristic that lowers the dependent variable-ie: a car with a negative residual is either a good deal or has something wrong with it

(ie: it’s not a Ford)

6.4 Residual Analysis-a positive residual can indicate an observation is overvalued or has an

unmeasured characteristic that increases the dependent variable-ie: a hockey team with a positive residual is either playing very well or

has an unmeasured positive factor (ie: it’s from Edmonton, city of champions)

6.4 Predicting with Logs-It is very common to express the dependent variable of a regression using natural logs, producing the true and estimated equations:

kk

kk

xxxy

uxxxy

ˆ...ˆˆˆgol

...log

22110

22110

-Where xj can also be in log form

-It may seem natural to predict y as:yey golˆ

-But this UNDERESTIMATES yhat

6.4 Predicting with Logs-From our 6 CLM assumptions, it can be shown that:

kk xxxeeXyE ...2/ 221102

)|(-Which gives us the simple prediction

adjustment:

yeey gol2/2

ˆ -Which is consistent, but not unbiased, and depends

highly on the error term having a normal distribution (MLR.6)

6.4 Predicting with Logs-Since in large samples normality isn’t required, forsaking MLR.6 we have

kk xxxeXyE ...0

22110)|(-where α0=E(eu)

-if we can estimate α0,

yey gol0ˆˆ

6.4 Predicting with Logs-In order to estimate α0hat,

1) Regress logy on all x’s and obtain logyhati’s:

2) For each observation, create mihat=elogyhati

3) Regress y on mihat without an intercept:

ikkiii

kk

xxxy

uxxxy

ˆ...ˆˆˆgol

...log

22110

22110

imy ˆ4) The coefficient on mihat is used to

predict y

6.4 Comparing R2’sBy definition, the R2 from the regression

kk xxxy ˆ...ˆˆˆˆ 22110 Is simply the squared correlation between the yi and the yihat.

This can be compared to the square of the sample correlation for the regression

imy ˆIn order to compare R2’s between a linear and log model.