Question

Provide definitions/explanations for the following (Use Equations Where Relevant)

Stochastic autoregressive process of order 2

a macroeconomic time series integrated of order 2, an I(2) time series

Answer 1

1. An autoregressive model for univariate, one‐dimensional, nonstationary, Gaussian random processes with evolutionary power spectra is introduced. At the same time, an efficient technique for numerically generating sample functions of such nonstationary processes is developed. The technique uses a recursive equation which: (1) Reflects the nature of the nonstationarity of the process whose sample functions are to be generated; and (2) involves a normalized univariate, one‐dimensional white noise sequence. The coefficients of the recursive equation are determined using the autocorrelation function of the process, which in turn is calculated from the evolutionary power spectrum at every time instant. Using the recursive equation with those coefficients, sample functions over a specified domain can be generated with substantial computational ease. Univariate, one‐dimensional, nonstationary processes with three different forms of the evolutionary power spectrum are modeled, and their sample functions are generated with the aid of an 11/750 VAX/VMS computer. The results indicate that the sample functions generated by the method presented reflect the prescribed probabilistic characteristics extremely well. This is seen from the closeness between the analytically prescribed autocorrelation functions and the corresponding sample autocorrelation functions computed from the generated sample functions.

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. An extension of CAR model is proposed in this article where the selection of the neighborhood depends on unknown parameter(s). This extension is called a Stochastic Neighborhood CAR (SNCAR) model. The resulting model shows flexibility in accurately estimating covariance structures for data generated from a variety of spatial covariance models. Specific examples are illustrated using data generated from some common spatial covariance functions as well as real data concerning radioactive contamination of the soil in Switzerland after the Chernobyl accident.

Estimation of AR parameters[edit]

The above equations (the Yule–Walker equations) provide several routes to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values.[6] Some of these variants can be described as follows:

A) Estimation of autocovariances or autocorrelations. Here each of these terms is estimated separately, using conventional estimates. There are different ways of doing this and the choice between these affects the properties of the estimation scheme. For example, negative estimates of the variance can be produced by some choices.

B) Formulation as a least squares regression problem in which an ordinary least squares prediction problem is constructed, basing prediction of values of Xt on the p previous values of the same series. This can be thought of as a forward-prediction scheme. The normal equations for this problem can be seen to correspond to an approximation of the matrix form of the Yule–Walker equations in which each appearance of an autocovariance of the same lag is replaced by a slightly different estimate.

C) Formulation as an extended form of ordinary least squares prediction problem. Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward prediction equations, relating to the backward representation of the AR model:

2. The time series perspective—cycles being determined by various random shocks which are propagated throughout the economy over time—is central to how modern macroeconomists now view economic fluctuations. VARs are a very common framework for modelling macroeconomic dynamics and the effects of shocks.

Macroeconomists tend to break series into a “non-stationary” long-run trend and a “stationary” cyclical component. “Business cycle analysis” relates to this modelling and explaining the cyclical components of the major macroeconomic variables. Fine in theory, but how is this done in practice? Simplest method: Log-linear trend I Estimated from regression log(Yt) = yt = α + gt + t I Trend component α + gt. I Zero-mean stationary cyclical component t . I Log-difference ∆yt (equivalent to growth rate) has two components: Constant trend growth g and the change in cyclical component ∆t .

National Bureau of Economic Research. Macroeconomic Time Series for the United States, United Kingdom, Germany, and France. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor]

The course will have three parts: 1 Time Series as a Framework for Modern Macro: We will discuss how time series provides a way to think about empirical macro, focusing particularly on Vector Autoregressions which are popular econometric models for forecasting and “what if?” scenario analysis. 2 Dynamic Stochastic General Equilibrium (DSGE) Models: Theoretically-founded models. We will cover the methods used to derive these models and simulate them on a computer. We will start with Real Business Cycle models and then move on to New-Keynesian models. 3 Financial Markets, Banking and Systemic Risk: We will cover risk spreads, credit rationing, financial intermediation, bank runs, banking regulation, systemic risk and bank balance sheet adjustments.