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

(a) Principle of least squares and fitting of the trend y = a bᵗ c^(t²)

Principle: the best-fitting curve is the one for which the sum of squares of the deviations between the observed values and the fitted values is least. Squaring prevents positive and negative errors cancelling and penalises large errors. Differentiating the sum of squares with respect to each parameter and equating to zero gives the normal equations. Use in trend fitting: take time t as the independent variable and the series as the dependent variable, choose a trend form (straight line, parabola, exponential) and estimate its constants by least squares. Fitting y = a bᵗ c^(t²): Step 1: Take logarithms: log y = log a + t log b + t² log c, that is Y = A + Bt + Ct², with Y = log y, A = log a, B = log b and C = log c. Step 2: This is a parabola in the logarithms, so least squares gives the normal equations ΣY = nA + BΣt + CΣt², ΣtY = AΣt + BΣt² + CΣt³ and Σt²Y = AΣt² + BΣt³ + CΣt⁴. Step 3: Measuring t from the middle of the series makes Σt = Σt³ = 0, so A = (ΣY − CΣt²)/n, B = ΣtY/Σt² and C = [nΣt²Y − Σt²ΣY]/[nΣt⁴ − (Σt²)²]. Step 4: Recover a = antilog A, b = antilog B and c = antilog C.

(b) Stationary time series: definition, testing, importance and the three models

Definition: a series is (weakly) stationary if its mean and variance are constant and its autocovariance between Yₜ and Yₜ₊ₖ depends only on the lag k, not on time t. Testing: plot the series and its correlogram (autocorrelations of a stationary series die out quickly, those of a non-stationary one decay slowly), and apply a unit-root test such as the Dickey-Fuller test of H₀: δ = 1 in ΔYₜ = (δ − 1)Yₜ₋₁ + Uₜ. Importance: regressions between non-stationary series can be spurious (high R² with no real relation), the usual t and F tests become invalid, and a stationary series has stable relationships that can be forecast. Checking the models (Uₜ is white noise with variance σ²): (i) Yₜ = Yₜ₋₁ + Uₜ is a random walk: Yₜ = Y₀ + ΣUᵢ, so Var(Yₜ) = tσ², which grows with time. It is non-stationary. (ii) Yₜ = δ + Yₜ₋₁ + Uₜ is a random walk with drift: Yₜ = Y₀ + δt + ΣUᵢ. Both the mean δt and the variance tσ² change with t. It is non-stationary. (iii) Yₜ = δYₜ₋₁ + Uₜ is a first-order autoregression. For |δ| < 1, E(Yₜ) = 0, Var(Yₜ) = σ²/(1 − δ²) and the autocovariance is δᵏσ²/(1 − δ²), all independent of t. It is stationary when |δ| < 1; at δ = ±1 it has a unit root and is non-stationary.

(c) Detecting heteroscedasticity and the Goldfeld-Quandt test

Methods: plotting residuals against fitted values or a regressor; Park test; Glejser test; Spearman's rank correlation test; Goldfeld-Quandt test; Breusch-Pagan-Godfrey test; White's test. Goldfeld-Quandt test: order the observations by the variable suspected of causing the changing variance, omit c middle observations, fit separate regressions to the first and last groups, and compute F = RSS₂/RSS₁ with the larger RSS in the numerator, comparing it with F(n₂ − k, n₁ − k) degrees of freedom. Assumptions: errors are normally distributed and independent (no autocorrelation); the model is correctly specified; the error variance changes monotonically with the ordering variable; the number of observations omitted is small (about one-third). Application: H₀: equal error variance; H₁: the variance rises with income. Step 1: RSS₁ = 377·17 and RSS₂ = 1536·8, each with 11 degrees of freedom. Step 2: F = 1536·8/377·17 = 4·07. Step 3: The 5% critical value F(11, 11) is 2·82 and the 1% value is 4·46. Since 4·07 > 2·82, H₀ is rejected at 5%; since 4·07 < 4·46, it is not rejected at 1%. Conclusion: there is heteroscedasticity, with error variance rising with income, significant at the 5% level but not at the 1% level.