The Heteroskedastic Consequences of an Arbitrary Variance...

The Heteroskedastic Consequences of an Arbitrary Variance for the Initial Disturbance of an AR(1) Model. This is based on Baltagi and Li (1990, 1992). Consider a simple AR(1) model $$ u_t=\rho u_{t-1}+\epsilon_t \quad t=1,2, \ldots, T \quad|\rho|<1 $$ with $\epsilon_t \sim \operatorname{IID}\left(0, \sigma_\epsilon^2\right)$ independent of $u_0 \sim\left(0, \sigma_\epsilon^2 / \tau\right)$, and $\tau$ is an arbitrary positive parameter. (a) Show that this arbitrary variance on the initial disturbance $u_0$ renders the disturbances, in general, heteroskedastic. (b) Show that $\operatorname{var}\left(u_t\right)=\sigma_t^2$ is increasing if $\tau>\left(1-\rho^2\right)$ and decreasing if $\tau<\left(1-\rho^2\right)$. When is the process homoskedastic? (c) Show that $\operatorname{cov}\left(u_t, u_{t-s}\right)=\rho^s \sigma_{t-s}^2$ for $t \geq s$. Hint: See the solution by Kim (1991). (d) Consider the simple regression model $$ y_t=\beta x_t+u_t \quad t=1,2 \ldots, T $$ with $u_t$ following the AR(1) process described above. Consider the common case where $\rho>0$ and the $x_t$ 's are positively autocorrelated. For this case, it is a standard result that the $\operatorname{var}\left(\widehat{\beta}_{O L S}\right)$ is understated under the stationary case (i.e., $\left.\left(1-\rho^2\right)=\tau\right)$, see problem 8 . This means that OLS rejects too often the hypothesis $H_0 ; \beta=0$. Show that OLS will reject more often than the stationary case if $\tau<1-\rho^2$ and less often than the stationary case if $\tau>\left(1-\rho^2\right)$. Hint: See the solution by Koning (1992).

The Heteroskedastic Consequences of an Arbitrary Variance for the Initial Disturbance of an AR(1) Model. This is based on Baltagi and Li (1990, 1992). Consider a simple AR(1) model
$$
u_t=\rho u_{t-1}+\epsilon_t \quad t=1,2, \ldots, T \quad|\rho|<1
$$
with $\epsilon_t \sim \operatorname{IID}\left(0, \sigma_\epsilon^2\right)$ independent of $u_0 \sim\left(0, \sigma_\epsilon^2 / \tau\right)$, and $\tau$ is an arbitrary positive parameter.
(a) Show that this arbitrary variance on the initial disturbance $u_0$ renders the disturbances, in general, heteroskedastic.
(b) Show that $\operatorname{var}\left(u_t\right)=\sigma_t^2$ is increasing if $\tau>\left(1-\rho^2\right)$ and decreasing if $\tau<\left(1-\rho^2\right)$. When is the process homoskedastic?
(c) Show that $\operatorname{cov}\left(u_t, u_{t-s}\right)=\rho^s \sigma_{t-s}^2$ for $t \geq s$. Hint: See the solution by Kim (1991).
(d) Consider the simple regression model
$$
y_t=\beta x_t+u_t \quad t=1,2 \ldots, T
$$
with $u_t$ following the AR(1) process described above. Consider the common case where $\rho>0$ and the $x_t$ 's are positively autocorrelated. For this case, it is a standard result that the $\operatorname{var}\left(\widehat{\beta}_{O L S}\right)$ is understated under the stationary case (i.e., $\left.\left(1-\rho^2\right)=\tau\right)$, see problem 8 . This means that OLS rejects too often the hypothesis $H_0 ; \beta=0$. Show that OLS will reject more often than the stationary case if $\tau<1-\rho^2$ and less often than the stationary case if $\tau>\left(1-\rho^2\right)$. Hint: See the solution by Koning (1992).

Key Concepts

Heteroskedasticity in Time Series

Heteroskedasticity refers to the phenomenon where the variance of the error terms or disturbances in a time series model is not constant over time. In the context of AR(1) models, an arbitrary variance specified for the initial disturbance can cause the subsequent disturbances to have time-dependent variance, thereby violating the homoskedasticity assumption that is often needed for standard inference procedures.

Autoregressive (AR) Process

An autoregressive process, particularly the AR(1) model, is a time series model in which the current value depends linearly on its immediately preceding value and a stochastic error term. This subject is central to the analysis, as the recursive structure of the AR(1) model means that any nonstandard properties in the initial condition can propagate through time and affect the overall behavior of the process.

Stationarity and Initialization Variance

Stationarity in an AR(1) process typically requires that the expected variance is constant across time. However, when the initial disturbance is allowed an arbitrary variance, the resulting variance of the process can be increasing or decreasing over time, depending on the relationship between the specified parameter and the process parameters. This concept explores how departures from the standard stationary initialization cause deviations from homoskedasticity.

Autocovariance Structure

The autocovariance function in an AR(1) process describes how observations at different points in time are correlated with each other. The structure is influenced by the autoregressive parameter, and understanding this relationship is crucial to analyzing how past values and variances affect current observations. The ability to express the covariance between terms in the process is key to understanding the temporal dependencies in the model.

Impact on OLS Estimation

In the presence of heteroskedastic disturbances, ordinary least squares (OLS) estimation can lead to understated or overstated standard errors, thereby affecting hypothesis testing and inference. This concept encompasses the implications for regression analysis when the error terms deviate from the homoskedastic assumption, illustrating how autocorrelation and time-varying variances influence the bias in estimated variances of the coefficients.

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