ECON4916 Chap.3 Regression Inference and Linear Restrictions
Regression Inference and Linear Restrictions
Regression analysis combines a fitting criterion with assumptions that support estimation and inference. This chapter derives ordinary least squares from squared-residual minimisation, rewrites estimation error as a product of sample regressor moments and regressor-error moments, and uses that decomposition to locate the rank and orthogonality conditions behind consistency.
It then explains why asymptotic normality adds a convergence rate and testing distribution rather than merely repeating consistency. Linear economic claims are translated into R beta equals r and assessed with a covariance-aware Wald statistic. The chapter closes by separating statistical evidence, economic magnitude, causality and forecast performance.
That separation matters because a small p-value can reject a restriction without validating the whole model or proving an economic mechanism.
What this chapter covers
- 01
Linear regression and OLS
- 02
Fitted values and residuals
- 03
Consistency and orthogonality
- 04
Asymptotic normality
- 05
Linear restriction matrices
- 06
Wald test interpretation
Worked example · free
Consistency from the OLS decomposition
- +1Substitute y = X beta + epsilon into the OLS formula to obtain beta-hat minus beta = (X prime X)^(-1)X prime epsilon.
- +1Scale both sample sums by T so the error becomes the inverse sample second-moment matrix times the sample orthogonality moment.
- +1Apply the law of large numbers to obtain E[x_t x_t prime] and E[x_t epsilon_t] as probability limits.
- +1Use nonsingularity to ensure the population regressor-moment matrix has an inverse.
- +1Combine the limits to obtain the inverse matrix times zero, so beta-hat converges in probability to beta.
Key terms
- Ordinary Least Squares
- An estimator choosing coefficients that minimise the sample sum of squared regression residuals.
- Fitted Value
- The outcome predicted by the estimated regression function for a given regressor observation.
- Residual
- The observed outcome minus its fitted value, calculated after estimating the regression coefficients.
- Consistency
- Convergence in probability of an estimator toward the true parameter as sample size grows.
- Rank Condition
- The nonsingularity requirement that regressors provide enough independent variation to identify coefficients.
- Linear Restriction
- A coefficient claim expressible as a matrix equation of the form R beta equals r.
Regression Inference and Linear Restrictions FAQ
Why does OLS consistency need orthogonality?
The estimator error contains the sample average of x_t times epsilon_t. A law of large numbers takes that term toward its population expectation. If the expectation is zero, the component vanishes; if it is not, more observations do not remove the systematic relation between regressors and the error.
What does asymptotic normality add?
Consistency says estimation error shrinks but gives neither a rate nor an approximate testing distribution. Asymptotic normality describes root-sample-size scaled error with a nondegenerate normal limit. That result supports large-sample standard errors, confidence intervals and hypothesis tests when the accompanying regularity and covariance assumptions are satisfied.
How do I write a joint restriction?
Put one independent linear claim in each row of R and its null value in the corresponding element of r. For beta_1 = 0 and beta_2 = 0, use two selector rows. Read every row back into words before calculating to ensure the matrix tests the intended proposition.
Does rejecting a null prove the alternative theory?
No. Rejection says the observed restriction discrepancy is too large relative to estimated uncertainty under the maintained null and model assumptions. It does not prove causality, validate all regressors or establish economic importance. Interpret the tested coefficient combination and then assess magnitude and identification separately.
Exam move
Rebuild regression inference from the estimator decomposition rather than memorising isolated results. Begin with y = X beta + epsilon, substitute into the OLS formula and identify the sample matrix and sample orthogonality moment. Beside each step, write the assumption it needs: a time-series law of large numbers, E[x_t epsilon_t] equal to zero and an invertible population second-moment matrix.
Next, separate coefficient estimation from uncertainty estimation. Ask whether serial correlation or changing variance alters the covariance estimator, and remember that robust standard errors do not repair an inconsistent coefficient. For restrictions, practise translating verbal claims into R and r before seeing any numbers.
Include equality between coefficients, sums of coefficients and several coefficients jointly equal to zero. Calculate the number of independent rows and write the Wald reference distribution. Finish with interpretation drills that distinguish rejection, economic magnitude, causal identification and predictive value.
In MATLAB output, locate the coefficient vector, covariance estimate and restriction being tested rather than treating a printed p-value as a complete answer. A clear null sentence should appear beside every matrix calculation.
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