Economics 706 Preliminary Examination, August 2015
Professor Francis X. Diebold
There is just one question, with five parts. Good luck!
Consider an N -variable dynamic single-factor model:








ε1t
λ1
µ1
y1t
 . 
 . 
 . 
 .. 
 .  =  ..  +  ..  ft +  .. 
εN t
λN
µN
yN t
ft = φ1 ft−1 + φ2 ft−2 + ηt ,
where yit is an indicator of real activity (e.g., retail sales growth), ft is a latent common
factor, and all stochastic shocks are white noise with zero mean and constant finite
variance, independent at all “own” and “cross” leads and lags.
(1) What is E(ft ) ? What is E(yit ) ? Under what conditions is ft covariance stationary?
Under what conditions is yit covariance stationary? From this point onward, assume
that those conditions are satisfied unless explicitly stated otherwise.
(2) Calculate the cross-covariance function of yi and yj , γyi yj (τ ) . How is it related to
the autocovariance function of ft , γf (τ ) ? Suppose that autoregressive lag-operator
polynomial of f has a pair of complex conjugate roots. What does that imply for the
behavior of γf (τ ) , as τ → ∞ ? What does that imply for the behavior of the spectral
density of ft , ff (ω) , ω ∈ [0, π] ?
(3) Explain in detail how you would estimate the unconditional joint density of the
observed variables (y1t , ..., yN t )0 using a kernel smoother with a Gaussian kernel
function. Under what conditions is your estimator consistent? Is consistency likely to be
comforting if N =12 and T =300? Why or why not?
(4) Cast the system in state space form and display its measurement and transition
equations. What can you say about the eigenvalues of the coefficient matrix in the
transition equation? Assuming that all stochastic shocks are Gaussian, how would you
evaluate the Gaussian log likelihood via (a) a time-domain prediction-error
decomposition in conjunction with the Kalman filter, and (b) frequency-domain
methods. What are comparative merits of the two approaches?
(5) Assume instead that ηt follows a conditionally Gaussian GARCH(1,1) process. Write
down an explicit expression for the process. What conditions are needed to ensure
positivity of the unconditional variance? Positivity of the conditional variance?
Covariance stationarity? Assuming that those conditions are satisfied, is ηt iid? Serially
uncorrelated? Serially independent? Unconditionally Gaussian? How does the state
space representation of the system change?