3-21 The prf and the sre The sample regression function (SRF) is the relationship that has been estimated =+ and we also know that u,=y, -y We use the srf to infer likely values of the PrF We also want to know how good our estimates of a and B are
3-21 The PRF and the SRF • The sample regression function (SRF) is the relationship that has been estimated. • and we also know that . • We use the SRF to infer likely values of the PRF. • We also want to know how “good” our estimates of and are. yt xt ˆ ˆ = ˆ + t t t u ˆ = y − y ˆ
Linearity In order to use ols. we need a model which is linear in the parameters(a and B). It does not necessarily have to be linear in the variables(y and x) Linear in the parameters means that the parameters are not multiplied together, divided, squared or cubed etc. Some models can be transformed to linear ones by a suitable substitution or manipulation, e.g. the exponential regression mode Y=eesny=a+BInX+u Then let yln yt and xln Xt a+x
3-22 Linearity • In order to use OLS, we need a model which is linear in the parameters ( and ). It does not necessarily have to be linear in the variables ( y and x). • Linear in the parameters means that the parameters are not multiplied together, divided, squared or cubed etc. • Some models can be transformed to linear ones by a suitable substitution or manipulation, e.g. the exponential regression model • Then let yt =ln Yt and xt=ln Xt yt = + xt +ut t t t u Yt e Xt e Y X u t = ln = + ln +
3-2 Linear and non linear models This is known as the exponential regression model. here the coefficients can be interpreted as elasticities Similarly, if theory suggests that y and x should be inversely related B y,=a+=tut then the regression can be estimated using ols by substituting But some models are intrinsically non-linear, e.g. B y,=a+, fu
3-23 Linear and Non-linear Models • This is known as the exponential regression model. Here, the coefficients can be interpreted as elasticities. • Similarly, if theory suggests that y and x should be inversely related: then the regression can be estimated using OLS by substituting • But some models are intrinsically non-linear, e.g. t t t u x y = + + t t x z 1 = t t ut y = + x +
3-24 Estimator or estimate? Estimators are the formulae used to calculate the coeficients Estimates are the actual numerical values for the coefficients
3-24 Estimator or Estimate? • Estimators are the formulae used to calculate the coefficients • Estimates are the actual numerical values for the coefficients
3-25 5 The assumptions Underlying the Classical Linear regression Model We observe data for x, but since y, also depends on up we must be specific about how the u, are generated. We usually make the following set of assumptions about the ut (the unobservable error terms): Technical notation Interpretation E(u)=0 The errors have zero mean 2. Var(u)=o2 The variance of the errors is constant and finite over all values ofx 3.Cov(a,4)=0 The errors are statisticall independent of one another 4.Cov(,x)=0 No relationship between the error and corresponding x variate
3-25 5 The Assumptions Underlying the Classical Linear Regression Model • We observe data for xt , but since yt also depends on ut , we must be specific about how the ut are generated. • We usually make the following set of assumptions about the ut (the unobservable error terms): • Technical Notation Interpretation 1. E(ut ) = 0 The errors have zero mean 2. Var (ut ) = 2 The variance of the errors is constant and finite over all values of xt 3. Cov (ui , uj )=0 The errors are statistically independent of one another 4. Cov (ut , xt )=0 No relationship between the error and corresponding x variate