By Vance Martin

This booklet presents a basic framework for specifying, estimating, and trying out time sequence econometric types. precise emphasis is given to estimation by means of greatest chance, yet different equipment also are mentioned, together with quasi-maximum probability estimation, generalized approach to moments estimation, nonparametric estimation, and estimation by means of simulation. an immense good thing about adopting the main of extreme chance because the unifying framework for the ebook is that a number of the estimators and attempt facts proposed in econometrics could be derived inside a chance framework, thereby offering a coherent motor vehicle for knowing their houses and interrelationships. not like many present econometric textbooks, which deal more often than not with the theoretical houses of estimators and try out data via a theorem-proof presentation, this booklet squarely addresses implementation to supply direct conduits among the idea and utilized paintings.

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**Example text**

Use the estimate from part (c), to plot the distribution of the number of strikes and interpret this plot. 32 The Maximum Likelihood Principle (e) Compute a histogram of yt and comment on its consistency with the distribution of strike numbers estimated in part (d). S. per annum expressed in days, yt . Durations are assumed to be drawn from an exponential distribution with unknown parameter θ y 1 exp − . θ θ Write the log-likelihood function for a sample of T observations. Derive and interpret the maximum likelihood estimator of θ.

4yt−1 ut−1 + 2zt , zt ∼ iid N (0, 1) . 8wt . 9u2t−1 zt ∼ iid N (0, 1) . m A sample of T = 4 observations, yt = {6, 2, 3, 1}, is drawn from the Poisson distribution θ y exp[−θ] f (y; θ) = . y! (a) (b) (c) (d) (e) Write the log-likelihood function, ln LT (θ). Derive and interpret the maximum likelihood estimator, θ. Compute the maximum likelihood estimate, θ. Compute the log-likelihood function at θ for each observation. Compute the value of the log-likelihood function at θ. 30 The Maximum Likelihood Principle (f) Compute gt (θ) = d ln lt (θ) dθ and ht (θ) = θ=θ d2 ln lt (θ) dθ 2 , θ=θ for each observation.

Conversely, the distribution is dependent if yt depends on its own lagged values and non-identical if it changes over time. 2 Count Model Consider a time series of counts modelled as a series of draws from a Poisson distribution f (y; θ) = θ y exp[−θ] , y! y = 0, 1, 2, · · · , where θ > 0 is an unknown parameter. 1 for θ = 2. By assumption, this distribution is the same at each point in time. In contrast to the data in the previous example where the random variable is continuous, the data here are discrete as they are positive integers that measure counts.