Exam Graph Drill

Econometric model Y = β₀ + β₁X + u: households with the same income consume different amounts, so the points scatter around the line. The vertical distance from a point to the line is the disturbance u (above the line u > 0, below u < 0).

Gujarati's 60 families (Table 2.1). At each income X there is a whole distribution of consumption Y. Its average is the conditional mean E(Y | Xᵢ) (big dots: 65, 77, …, 173). The line joining the conditional means is the population regression function, PRF: E(Y | Xᵢ) = 17 + 0.6Xᵢ.

Gujarati's figure 2.5. For one observation A at Xᵢ: in terms of the PRF, Yᵢ = E(Y | Xᵢ) + uᵢ (uᵢ = vertical gap to the PRF); in terms of the SRF, Yᵢ = Ŷᵢ + eᵢ (eᵢ = vertical gap to the SRF). Here the SRF underestimates the true E(Y | Xᵢ), so eᵢ > uᵢ. Data example: A = (240, 189) gives u = 189 − 161 = 28 and, with Ŷ = 24.45 + 0.509X, e = 189 − 146.65 = 42.35.

Two random samples of 10 families from the same population (Tables 2.4 and 2.5) give two different SRFs: Ŷ = 24.45 + 0.509X and Ŷ = 17.17 + 0.576X. Neither is the true PRF; we can never be sure which is closer. That is sampling fluctuation.

Sample 1 (Table 2.4) with the OLS line Ŷ = 24.45 + 0.509X. Each residual eᵢ = Yᵢ − Ŷᵢ is the vertical distance from the point to the line: green above the line (e > 0), red below (e < 0). Positive and negative residuals cancel: Σeᵢ = 0.

For each observation the total deviation from the mean, Yᵢ − Ȳ, splits into the part the line explains, Ŷᵢ − Ȳ, plus the residual eᵢ. Square and sum over all i: TSS = ESS + RSS. Data example: Yᵢ = 155 at Xᵢ = 240 gives 44 = 35.65 + 8.36; for the whole sample 8890 = 8552.73 + 337.27.

Homoscedasticity (equal spread): Var(uᵢ | Xᵢ) = σ². The conditional distributions have the SAME width at every X, so the points form a band of constant thickness around the PRF.

Positive autocorrelation (violation): a positive u tends to be followed by another positive u, so the series moves in long runs above and below zero. (Negative autocorrelation would flip sign almost every period.)

Repeat the sampling many times and b₂ takes a different value each time: it has a sampling distribution. OLS is unbiased: its distribution is centred on the true β₂, E(b₂) = β₂. A biased estimator β̃₂ is centred somewhere else; the gap is the bias.

Overlap (shaded) = the part of Y's variation explained by X (ESS). The rest of the Y circle is unexplained (RSS). r² = shaded area / whole Y circle = ESS/TSS.

H₀: β₂ = c vs H₁: β₂ ≠ c at α = 5%, df = n − 2 = 8. Put α/2 = 0.025 in EACH tail: critical values ±2.306. If the computed t falls in a red tail (|t| > 2.306), reject H₀; in the middle 95% region, do not reject.

The 95% interval for β₂ is 0.509 ± 2.306 × 0.035 = [0.428, 0.590]. The confidence-interval approach (two-sided only): c = 0.55 lies inside → do not reject; c = 0.60 lies outside → reject. 0 is far outside → slope significant.

95% confidence band for E(Y|X₀): Ŷ₀ ± t₀.₀₂₅ × se(Ŷ₀). Because se(Ŷ₀) has (X₀ − X̄)², the band is narrowest at the mean income X̄ = 170 (se = 2.05) and widens hyperbolically as X departs from X̄.

Reciprocal model CM = 81.79 + 27,273(1/PGNP). As per capita GNP increases indefinitely, 1/PGNP → 0 and child mortality falls toward horizontal asymptote β₁ ≈ 82 deaths per 1000. Slope = −β₂/X² < 0.

Logarithmic reciprocal model ln Y = β₁ − β₂(1/X), or Y = exp(β₁ − β₂/X). At first Y increases at an increasing rate, reaches an inflection point at X = β₂/2, then increases at a decreasing rate toward the asymptote exp(β₁). Models short-run production.

Expenditure on services 1993:1-1998:3. Linear trend: Ŷ = 2405.85 + 19.69t (constant $19.69B per quarter). Log-lin model: Ŷ = 2413.90 × e^(0.00743t) (constant quarterly growth 0.743%, compound 0.746%).