By Vance Martin

This e-book offers a normal framework for specifying, estimating, and checking out time sequence econometric types. certain 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 way of simulation. a major good thing about adopting the primary of utmost probability because the unifying framework for the ebook is that a number of the estimators and try out facts proposed in econometrics should be derived inside a probability framework, thereby offering a coherent motor vehicle for realizing their homes and interrelationships. unlike many latest econometric textbooks, which deal often with the theoretical homes of estimators and try out information via a theorem-proof presentation, this publication squarely addresses implementation to supply direct conduits among the speculation and utilized paintings.

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**Extra resources for Econometric Modelling with Time Series: Specification, Estimation and Testing**

**Example text**

In a large number of cases, this may be achieved using standard calculus. Chapter 3 discusses numerical approaches to the problem of finding maximum likelihood estimates when no analytical solutions exist, or are difficult to derive. 10 Poisson Distribution Let {y1 , y2 , · · · , yT } be iid observations from a Poisson distribution f (y; θ) = θ y exp[−θ] , y! where θ > 0. y2 ! ) . T Consider the following T = 3 observations, yt = {8, 3, 4}. 191 . 3 for values of θ ranging from 0 to 10. Even though the Poisson distribution is a discrete distribution in terms of the random variable y, the log-likelihood function is continuous in the unknown parameter θ.

01 . 893 To avoid this problem, alternative models of interest rates are specified where the stationary distribution is just defined over the positive region. A well known example is the CIR interest rate model (Cox, Ingersoll and Ross, 1985) which is discussed in Chapters 2, 3 and 12. 21) where θ = α, ρ, σ 2 and the substitution ρ = 1+β is made for convenience. 13. 21) is 1 1 1 ln LT (θ) = − ln 2π − ln σ 2 − 2 2 2 2σ (T − 1) T t=2 (rt − α − ρrt−1 )2 , where the sample size is reduced by one observation as a result of the lagged term rt−1 .

5 is T f (y1 , y2 , · · · , yT ; θ) = f (y1 ; θ) t=2 f (yt |yt−1 ; θ) , where the conditional distribution is 1 (yt − ρyt−1 )2 f (yt |yt−1 ; θ) = √ exp − 2σ 2 2πσ 2 , 12 The Maximum Likelihood Principle and the marginal distribution is f (y1 ; θ) = 1 2πσ 2 / (1 − ρ2 ) exp − y12 . 2σ 2 / (1 − ρ2 ) Non-stochastic explanatory variables In the case of non-stochastic explanatory variables, because xt is deterministic its probability mass is degenerate. Explanatory variables of this form are also referred to as fixed in repeated samples.