A spurious relationship occurs when two independent variables are incorrectly identified as being related. A simple test of independence is based on the estimated correlation coefficient, \(\widehat{ho}\).
(a) Consider the following bivariate models

in which \(v_{1 t}, v_{2 t}\) are iid \(N\left(0, \sigma^{2}\right)\) with \(\sigma^{2}=1\). Simulate each bivariate model 10000 times for a sample of size \(T=100\) and compute the correlation coefficient, \(\hat{ho}\), of each draw. Compute the sampling distributions of \(\widehat{ho}\) for the four sets of bivariate models and discuss the properties of these distributions in the context of the spurious regression problem.
(b) Repeat part (a) with \(T=500\). What do you conclude?
(c) Repeat part (a), except for each draw estimate the regression model
\[
y_{2 t}=\beta_{0}+\beta_{1} y_{1 t}+u_{t}, \quad u_{t} \sim \operatorname{iid}\left(0, \sigma^{2}\right)
\]
Compute the sampling distributions of the least squares estimator \(\widehat{\beta}_{1}\) and its \(\mathrm{t}\) statistic for the four sets of bivariate models. Discuss the properties of these distributions in the context of the spurious regression problem.

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