(a) Explain the principle of least squares. How it is used in fitting trend in time series analysis ? Explain the fitting of trend for the curve y=ab^tc^t². 15 marks

(b) Define stationary time series. How would you test the stationarity of the given

Question

(a) Explain the principle of least squares. How it is used in fitting trend in time series analysis ? Explain the fitting of trend for the curve y=ab^tc^t². 15 marks

(b) Define stationary time series. How would you test the stationarity of the given time series ? Write the importance of stationary time series. Check the following time series for stationarity.

(i) Yₜ = Yₜ₋₁ + Uₜ

(ii) Yₜ = δ + Yₜ₋₁ + Uₜ

(iii) Yₜ = δ Yₜ₋₁ + Uₜ ; -1 ≤ δ ≤ 1 15 marks

(c) State the different methods of detecting the presence of heteroscedasticity. Explain in brief the Goldfeld-Quandt Test for detecting the presence of heteroscedasticity. Also write the assumption required to apply this test.

For a data on consumption expenditure in relation to income for a cross section of 30 families, after dropping the middle 4 observations, the OLS regression based on the first 13 and the last 13 observations and their associated residual sum of squares are as follows :

Regression based on the first 13 observations :

Ŷᵢ = 3.4094 + 0.6968 Xᵢ

(r² = 0.8887, RSS₁ = 377.17, df = 11)

Regression based on the last 13 observations :

Ŷᵢ = -28.0272 + 0.7941 Xᵢ

(r² = 0.7681, RSS₂ = 1536.8, df = 11)

Check the presence of heteroscedasticity for the above given results and write your conclusion.

(F_(11, 11, 5%) = 2.82, F_(11, 11, 1%) = 4.46, F_(13, 13, 5%) = 2.53, F_(13, 13, 1%) = 3.82) 20 marks

UPSC Answer Check · Accepted Answer

(a) Principle of Least Squares and Trend Fitting The principle of least squares states that the best-fitting curve to a set of observations is the one for which the sum of the squares of the vertical deviations (residuals) between observed and estimated values is minimum, i.e. minimise Σ(Yᵢ − Ŷᵢ)². Squaring removes sign cancellation and penalises large deviations heavily; differentiating this sum with respect to each parameter and equating to zero yields the normal equations, whose solution gives the least-squares estimators. In time series, trend is fitted by treating time t as the independent variable and the series Yₜ as dependent, choosing a functional form (linear, quadratic, exponential, logistic) that reflects the underlying growth mechanism. For the curve y = abᵗc^t², take logarithms: log y = log a + t log b + t² log c, i.e. Y = A + Bt + Ct², where Y = log y, A = log a, B = log b, C = log c. This is linear in the transformed parameters, so the normal equations are: ΣY = nA + BΣt + CΣt² ΣtY = AΣt + BΣt² + CΣt³ Σt²Y = AΣt² + BΣt³ + CΣt⁴ Solving these three equations gives A, B, C, and back-transformation gives a = antilog A, b = antilog B, c = antilog C. The curve captures exponential growth modified by a squared-exponential term, useful where growth accelerates or decelerates over time. (b) Stationarity A time series is (weakly/covariately) stationary if its mean E(Yₜ) = μ, variance Var(Yₜ) = σ², and autocovariance Cov(Yₜ, Yₜ₊ₖ) = γₖ are all finite and constant over time, independent of t; strong stationarity requires the entire joint distribution to be time-invariant. Stationarity is tested graphically (ACF/PACF plots), by unit root tests (Dickey–Fuller, ADF, Phillips–Perron), or by the correlogram test. Its importance lies in valid inference: non-stationary series give spurious regressions, invalid t and F statistics, and unreliable forecasts. (i) Yₜ = Yₜ₋₁ + Uₜ is a pure random walk. Substituting recursively, Yₜ = Y₀ + ΣUᵢ, so Var(Yₜ) = tσ² grows with t — variance is non-constant, hence non-stationary. (ii) Yₜ = δ + Yₜ₋₁ + Uₜ is a random walk with drift. Yₜ = Y₀ + δt + ΣUᵢ; the mean grows linearly with t and variance grows with t, so it is non-stationary. (iii) Yₜ = δYₜ₋₁ + Uₜ is an AR(1) process. Its mean is zero, variance is σ²/(1−δ²) and autocovariance γₖ = δᵏσ²/(1−δ²), both finite and time-invariant provided |δ| < 1. Hence it is stationary for −1 < δ < 1; at δ = ±1 it reduces to a random walk and becomes non-stationary. (c) Detecting Heteroscedasticity Methods include: graphical inspection of residuals against fitted values or an explanatory variable; Park test; Glejser test; Spearman's rank correlation test; Goldfeld–Quandt test; White's general test; and Breusch–Pagan–Godfrey test. The Goldfeld–Quandt test proceeds by ordering observations by the variable suspected to cause heteroscedasticity, omitting c central observations, fitting separate regressions to the first n₁ and last n₂ observations, and computing F = RSS₂/RSS₁ (larger over smaller), compared with F(n₂−k, n₁−k) at the chosen level. Assumptions: errors are normal; under H₀ (homoscedasticity) variance is constant; the alternative is that variance increases (or decreases) monotonically with the ordering variable; the two sub-samples are independent. Here, H₀: homoscedasticity; H₁: variance increases with income. F = RSS₂/RSS₁ = 1536.8/377.17 = 4.075, with df (11, 11). Since 4.075 > F₀.₀₅(11,11) = 2.82, we reject H₀ at 5%; but 4.075 < F₀.₀₁(11,11) = 4.46, so it is not significant at 1%. Since RSS₂ > RSS₁, variance rises with income. Conclusion: heteroscedasticity is present in consumption expenditure, significant at the 5% level.

(a) Explain the principle of least squares. How it is used in fitting trend in time series analysis ? Explain the fitting of…

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