Question

During a meeting in which you brief your supervisor on a forecasting model, you mention Akaike Information Criterion (AIC) and Schwartz Bayesian Criterion (SBC). The supervisor is familiar with the concept of R2 (coefficient of determination) as the proportion of variance in the data that is explained by the model. Briefly explain why you put more emphasis on AIC and SBC when you decide between models. ALSO.

Explain the Markovian Property, in words, to that same supervisor.

The fundamental matrix of a Markov Model describes the behavior of the system in steady-state (T/F - explain).

Answer 1

MORE EMPHASIS ON AIC AND SBC TO DECIDE BETWEEN MODELS:

   We know that R square is the explained variance in response variables by the predictors. Or in other words it is the information captured from the data by the model.

Though it is necessary to measure the information captured, it is very important to find out the information lost from the data inorder to improve the model performance and get better predictions. Thus AIC and SBC comes into play.

These two measures are tradeoff between model goodness of fit and complexity that helps to reduce the estimated or forecasted information lost.

Calculating the correct order p of autoregressive process is important. One approach to choose the appropriate p is to perform hypothesis test on final lag. For example, say if we start from estimating AR(4) modeland test whether the coefficient on the fourth lag is significant at appropriate significance level say 5%; is it is not significant we drop it and estimate AR(3) model and so on.

An alternative approach is to estimate p by using the INFORMATION CRITERION like AKAIKE INFORMATION CRITERION and SCHWARTZ BAYESIAN INFORMATION CRITERIA.

AIC is given by,

where T is the number of observations used for estimation and k is the number of predictors in the model. The idea here is to penalise the fit of the model (SSE) with the number of parameters that needs to be estimated. The modle with minimum AIC is the best model for forecasting.

A related measure is the schwartz BIC given by, .

As with AIC, minimizing BIC is intended to give the best model. The model chosen by BIC is either the same as that chosen by AIC, or one with fewer parameters. Because BIC penalises the number of parameters more heavily than AIC.

While R square is widely used, its tendency to select too many predictor variables makes it less suitable for forecasting.Thus AIC or SBC is preferred because it has the feature that is given a true underlying model, this select that model given enough data.

MARKOVIAN PROPERTY:

Markov property says that whatever happens next in the process only depend on how it is right now(present state). It doesnt have a memory of how it was before.

Basically you dont need past states to do a optimal decision, all you need is the current state.This is because you could encode everything you need from the past on your current state to do a good decision. Still history matters...

F​UNDAMENTAL MATRIX OF MARKOV MODEL DO NOT DESCRIBE THE BEHAVIOUR OF STEADY STATE SYSTEM:

If we consider a markov system, it will move from state to state for all time periods and the average probabilities of moving from state to state for all periods will remain constant in long run.

Thus the average probabilities that the system will be in a certain state after a large number of transition periods are called STEADY STATE PROBABILITIES. This doesn't mean that the system stays in one state.

In a markov process, after a number of periods have passed, the probabilities will approach a steady state. Thus in future, the state probabilities become constant.

Thus future state probabilities can be computed from initial starting state probabilities from fundamental markov transition probability matrix. But the probability of ending up in a particular state in the future is not dependent on the starting state. Thus the given statement is false.

(i.e) "THE F​UNDAMENTAL MATRIX OF MARKOV MODEL DO NOT DESCRIBE THE BEHAVIOUR OF STEADY STATE SYSTEM".