4.6.1 Conditional vs. Unconditional Distributions

4.6.1  Conditional vs. Unconditional Distributions

Time series analysis is different from statistical estimation. Statistical estimators apply to realizations {x[1], … , x[m–1], x[m]} of samples {X[1], … , X[m–1], X[m]}. Time series analysis applies to realizations {–αx, … , –1x, 0x} of segments of stochastic processes {–αX, … , –1X, 0X}. A segment of a stochastic process is not a sample. It lacks an important property of samples. The random vectors X[k] that make up a sample are IID. The random vectors tX that make up a segment of a stochastic process may not be. This should be obvious from our Mt. Washington example. Temperatures follow trends. Cold days tend to follow cold days; warm days tend to follow warm days. Clearly, the random vectors tX in that example are not independent. They are also not identically distributed, since temperatures follow seasonal patterns.

Since terms tX of a process need not be independent, we must distinguish between terms’ conditional and unconditional distributions. The conditional distribution of tX as of time t – k is its distribution conditional on all information available at time t – k, but especially on realized values tkx, t–k–1x, t–k–2x, … Usually, we don’t need to know all preceding values. Only a handful of the most recent values may be relevant.

Consider a somewhat contrived process Y. All terms tY are equal,


and are U(0,1). Two realizations of the process are illustrated in Exhibit 4.7.

Exhibit 4.7: Terms of the process Y are equal and U(0,1). Two realizations are depicted.

In this example, 1Y has unconditional distribution U(0,1), but its distribution conditional on information at time 0 is degenerate, with 1Y = 0y.

We indicate the conditional expectation of a term tX as of time tk as t–kE(tX). We indicate the unconditional expectation as simply E(tX). Standard deviations, variances, skewnesses, and kurtoses are treated similarly. For example, the unconditional standard deviation of 1X is denoted std(1X). Conditional on information available at time 0, it is denoted 0std(1X).

Conditional parameters, such as a mean or standard deviation conditional on information available through time t – k, can also be indicated as t|tkμ or t|tkσ. Corresponding unconditional parameters are indicated tμ or tσ. Conditional or unconditional CDFs and PDFs are indicated similarly: t|tkΦ and t|tkϕ or tΦ and tϕ.

At the risk of seeming repetitive, let’s emphasize that the above conventions afford two means of specifying parameters of terms of stochastic processes. One employs operators such at E( ), std( ) or skew( ). The other employs standard symbols for parameters, such as μ, σ or η1The two approaches employ preceding superscripts differently. For example, the unconditional standard deviation of tX can be written as either std(tX) or tσ. Conditional on information available at time t – k, these become tkstd(tX) and t|tkσ. So, to give an even blunter example, tE( ) indicates a conditional expectation whereas tμ indicates an unconditional expectation.

In Chapter 3, we introduced shorthand notation for certain families of probability distributions. For example, we used notations N(μ,σ2) and U(a,b) to indicate univariate normal and uniform distributions. We used a tilde to indicate that a random variable had one of these distributions, so X ~ N(0,1) would mean that a random variable X was standard normal. We now extend that notation to terms of a stochastic process, leaving the tilde unmodified if a distribution is unconditional, or indicating some point in time above the tilde if the distribution is conditional on information available at that point in time. For example, tX ~ N(μ,σ2) indicates that term tX is unconditionally normal. indicates that term tX is uniformly distributed conditional on information available at time 0.