A3 Variances And Standard Errors Of Leastsquares Estimators

Now by the definition of variance, we can write var (&) = E[fr — E(h)]2

= E(p2 — p2)2 since E(fh) = p2 = E^^kWi^ using Eq. (4) above

= E(k2u2 + kirn, +-----+ k;iun + 2k1k2u1u2 +-----+ 2kn—1knun—1un)

102 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Since by assumption, E(u2) _ a2 for each i and E(uiuj) _ 0, i _ j, it follows that var (ft) _ a 2 £ k2

xi2 i

The variance of ft can be obtained following the same line of reasoning already given. Once the variances of ft and ft are obtained, their positive square roots give the corresponding standard errors.

3A.4 COVARIANCE BETWEEN ft AND ft

By definition, cov(ft, ft) _ E{[y01 - E(ft)][ft - E(ft)]}

where use is made of the fact that ft _ Y - ftX and E(ft) _ Y - j2X, giving j?1 - E(ft) _ -Xi(ji2 - ft)- Note: var^) is given in (3.3.1).

3A.5 THE LEAST-SQUARES ESTIMATOR OF a2

Recall that

Therefore,

Subtracting (10) from (9) gives yi _ ft Xi + (ui - u) (11)

Therefore, substituting (11) into (12) yields ui _ ftXi + (ui - u) - ftXi (13)

Gujarati: Basic I. Single-Equation 3. Two-Variable © The McGraw-Hill

Econometrics, Fourth Regression Models Regression Model: The Companies, 2004

Edition Problem of Estimation

CHAPTER THREE: TWO-VARIABLE REGRESSION MODEL 103

Collecting terms, squaring, and summing on both sides, we obtain

E u2 = (k — ¡2)2 Exi + E(Ui — uu)2 — 202 — ¡2) E x(Ui — u) (14) Taking expectations on both sides gives e(E = Ex2E^2 — ¡2)2 + E [E(u — u)2] — 2E [(¡32 — ¡2) Ex(u — U)

y^x2 var (jS2) + (n — 1) var(U;) — 2E [y^U(xiUi)j

where, in the last but one step, use is made of the definition of ki given in Eq. (3) and the relation given in Eq. (4). Also note that

22 nU

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