###### 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 {^{–α}** x**, … ,

^{–1}

**,**

*x*^{0}

**} of segments of stochastic processes {**

*x*^{–α}

**, … ,**

*X*^{–1}

**,**

*X*^{0}

**}. A segment of a stochastic process is**

*X**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

*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*

^{t}**X***in that example are not independent. They are also not identically distributed, since temperatures follow seasonal patterns.*

^{t}**X**Since terms * ^{t}X* of a process need not be independent, we must distinguish between terms’ conditional and unconditional distributions. The conditional distribution of

*as of time*

^{t}**X***t*–

*k*is its distribution conditional on all information available at time

*t*–

*k*, but especially on realized values

^{t}^{–k}

**,**

*x*

^{t–k}^{–1}

**,**

*x*

^{ t–k}^{–2}

**, … Usually, we don’t need to know all preceding values. Only a handful of the most recent values may be relevant.**

*x*Consider a somewhat contrived process ** Y**. All terms

*are equal,*

^{t}Y[4.43]

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

**are equal and**

*Y**U*(0,1). Two realizations are depicted.

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

We indicate the conditional expectation of a term * ^{t}X* as of time

*t*–

*k*as

*(*

^{t–k}E*). We indicate the unconditional expectation as simply*

^{t}X*E*(

*). Standard deviations, variances, skewnesses, and kurtoses are treated similarly. For example, the unconditional standard deviation of*

^{t}X^{1}

*X*is denoted

*std*(

^{1}

*X*). Conditional on information available at time 0, it is denoted

^{0}

*std*(

^{1}

*X*).

Conditional parameters, such as a mean or standard deviation conditional on information available through time *t* – *k*, can also be indicated as ^{t|t–k}μ or ^{t|t–k}σ. Corresponding unconditional parameters are indicated * ^{t}*μ or

*σ. Conditional or unconditional CDFs and PDFs are indicated similarly:*

^{t}

^{t}^{|t–k}Φ and

^{t}^{|t–k}ϕ or

^{t}Φ and

*ϕ.*

^{t}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 η_{1}. *The two approaches employ preceding superscripts differently.* For example, the unconditional standard deviation of * ^{t}X* can be written as either

*std*(

*) or*

^{t}X*σ. Conditional on information available at time*

^{t}*t*–

*k*, these become

^{t–k}*std*(

*) and*

^{t}X

^{t}^{|t–k}σ. So, to give an even blunter example,

*( ) indicates a conditional expectation whereas*

^{t}E*μ indicates an unconditional expectation.*

^{t}In Chapter 3, we introduced shorthand notation for certain families of probability distributions. For example, we used notations *N*(μ,σ^{2}) 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, * ^{t}X* ~

*N*(μ,σ

^{2}) indicates that term

*is unconditionally normal. indicates that term*

^{t}X*is uniformly distributed conditional on information available at time 0.*

^{t}X