(a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2, ...}, where eᵢ's are i.i.d. N(0, σ²), obtain the correlogram function. Find its form, when all the weights are equal and their sum

Question

(a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2, ...}, where eᵢ's are i.i.d. N(0, σ²), obtain the correlogram function. Find its form, when all the weights are equal and their sum is 1. (15 marks)

(b)

The marks obtained by student A in Mathematics and Language tests of maximum marks 150 each are 120 and 105 respectively. Find out in which subject, student A is more able as compared to other students based on the measure of T score. The following table gives a sample of marks obtained by 15 students of the same class :

Score in Mathematics | Score in Language ---|--- 100 | 67 75 | 63 88 | 73 85 | 77 92 | 60 94 | 53 93 | 50 84 | 48 67 | 38 96 | 73 100 | 36 102 | 45 94 | 47 73 | 39 83 | 56

15 marks

Score in Mathematics | Score in Language
---|---
100 | 67
75 | 63
88 | 73
85 | 77
92 | 60
94 | 53
93 | 50
84 | 48
67 | 38
96 | 73
100 | 36
102 | 45
94 | 47
73 | 39
83 | 56

(15 marks)

(c) Describe the 2-stage least squares (2SLS) method of estimation of parameters in linear regression model. Also, state the assumptions and discuss its properties. (20 marks)

UPSC Answer Check · Accepted Answer

(a) Time series and MA correlogram. A time series is an ordered sequence of random variables Yₜ, t=1,2,... observed over successive time points, whose joint distribution may exhibit dependence, trend and seasonality. It is analysed through its mean function, autocovariance function and correlogram. For an MA(m) process generated by iid N(0,σ²) shocks eₜ, write Yₜ = a₁ eₜ + a₂ eₜ₋₁+...+aₘ eₜ₋ₘ₊₁. Its mean is zero and autocovariance at lag k is γ(k)=Cov(Yₜ,Yₜ₊ₖ)=σ²∑ᵢ₌₁^m-k aᵢ aᵢ₊ₖ, for k=0,...,m-1; by symmetry γ(-k)=γ(k), and γ(k)=0 for |k|≥m because the moving-average window has finite length. The process is weakly stationary because its mean is constant and its autocovariance depends only on the lag. The normality of the shocks gives joint normality of the series, but the correlogram depends only on these second moments. The correlogram is ρ(k)=γ(k)/γ(0), so it has the same truncated triangular shape. If all weights are equal and sum to one, aᵢ=1/m, then γ(k)=σ²(m-k)/m² and γ(0)=σ²/m; hence ρ(k)=(m-|k|)/m for |k|<m and ρ(k)=0 for |k|≥m.

(b) T-score comparison. For the 15 students, the Mathematics scores sum to 1326, giving x̄_M=88.4; the Language scores sum to 825, giving x̄_L=55.0. The squared deviations sum to 1543.6 for Mathematics and 2514 for Language. Using the sample standard deviations, s_M=(1543.6/14)¹/2≈10.50 and s_L=(2514/14)¹/2≈13.40. The T-score is T=50+10(X−x̄)/s, which places the class mean at 50 and one standard deviation at 10. For student A, T_M=50+10(120−88.4)/10.50≈80.1, and T_L=50+10(105−55)/13.40≈87.3. If the class is treated as the population, the corresponding standard deviations are 10.14 and 12.95, giving T_M≈81.2 and T_L≈88.6. In either convention, A’s T-score is higher in Language, so A is relatively more able in Language compared with classmates, even though the absolute Mathematics mark is larger and the maximum marks in both subjects are the same.

(c) 2SLS. In a linear structural equation y=Xβ+u, some regressors in X may be endogenous because they are jointly determined with u. Let Z be a matrix of valid instruments, including all exogenous variables. 2SLS proceeds in two stages. First, each endogenous regressor is regressed on Z to obtain fitted values X̂=P_Z X, where P_Z=Z(Z'Z)⁻¹Z' is the projection matrix on the instrument space. This first stage is the reduced-form projection that replaces endogenous regressors with the part explained by instruments. Second, y is regressed on X̂; the estimator is β̂_2SLS=(X'P_Z X)⁻¹X'P_Z y. It is an IV estimator; when all regressors are exogenous, P_Z X = X and 2SLS reduces to OLS. The key assumptions are linearity in parameters; instrument exogeneity E(Z'u)=0; relevance, meaning E(Z'X) has full column rank so endogenous regressors are predicted with non-zero variation; no perfect multicollinearity among instruments; and finite moments sufficient for the central limit theorem. Under these assumptions, by the law of large numbers, plim β̂_2SLS=β because plim X'P_Z X/n converges to a positive definite matrix and plim X'P_Z u/n=0, so the probability limit of the normal equations is Qβ=0. By the continuous mapping theorem and a central limit theorem, √n(β̂_2SLS−β)→N(0,σ²Q⁻¹), where Q=plim(X'P_Z X/n). If σ² is estimated, it should be computed from the original structural residuals uᵢ=yᵢ−xᵢ'β̂_2SLS, not from the second-stage fitted residuals, to preserve the correct variance estimate. OLS is inconsistent when regressors are endogenous because plim X'u/n≠0; 2SLS removes this bias by replacing endogenous regressors with their instrument-based projections. In simultaneous-equation systems, 2SLS is applied equation by equation using instruments that are correlated with endogenous regressors but uncorrelated with the structural disturbance. In Indian agricultural policy, for example, supply response to output price may be endogenous because price and quantity are jointly determined; rainfall or transport cost can serve as instruments for price, making 2SLS useful for estimating causal supply elasticities.

(a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2, ...}…

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