Forecasting in Practice

Predictions about future events are inherent to all decisions.

In policy, business and finance decisions, uncertainty related to future outcomes represents risk.

  More informative predictions mean more optimal decisions and more efficient risk

management.

Forecasting is an approach to formulating predictions based on an observed historic data

sample.

A forecast uses rigorous methods to match the data to a pattern, then extrapolate the pattern beyond the end of the sample.

Forecasting in Practice

A good forecasting methodology:

  efficiently extracts useful information from the data when matching it to a pattern;   constructs a pattern that can be practically extrapolated into the future;

  appropriately quantifies all sources of uncertainty associated with the prediction.

Forecasting is based on the fundamental assumption that a pattern in the historic data remains valid in the future.

Forecasts are routinely used in a wide range of settings and for a variety of purposes; some examples:

  medium-term inflation forecasts for monetary policy;

  long-term GDP growth forecasts for development strategies;

 short-term orders forecasts for replenishment and supply chain management;

 short, medium and long-term returns forecasts for investment portfolio allocation.

Reserve Bank of Australia Economic Outlook Report, August 2021

GDP

Forecast scenarios, December 2019 = 100

index

110

100

90

80

index

110

100

90

80

2015           2017           2019           2021           2023

Sources: ABS; RBA

Stochastic Process

A stochastic process is a collection of random variables that are ordered in time.

A a stochastic process is also called a time series process:

  each observation is a random variable;

 observations evolve in time according to some probabilities;

 we will refer to the stochastic process as the underlying data generating process (DGP)

that generates the time-series data we observe.

A stochastic process is denoted by {yt  : t ∈ Z} or simply {yt}.

Realisations, Moments

A realisation is one of a (typically) ininite set of values of yt,t = 1, . . . ,T, randomly generated according to the probability distribution underlying the stochastic process.

Time-series data is a realisation of a stochastic process.

We are typically interested in the following moments characterizing the probability distribution:

Mean: µt ≡ E(yt), t = 1, . . . ,T ; which can be interpreted as the average value of yt taken over all possible realisations.

Variance: V0,t ≡ Var(yt) = E((yt− µt)2 ), t = 1, . . . ,T; i.e., the average of square deviations from the mean.

Covariance:Vk,t ≡ Cov(yt,yt−k) = E((yt− µt)(yt−k− µt−k)), t = k + 1, . . . ,T .

Stationarity

Several forms of stationarity that can be used to describe a stochastic process. We keep things simple with the following.

Definition

A stochastic process is stationary if and only if the mean, variance, and all covariances exist and are independent of time. Specifically, for all t,

E(yt) = µ,

Var(yt) = σy(2) = V0 ,

Cov(yt,yt−k) = Vk,        k ≥ 1,

  Stationarity is a property of the stochastic process.

  Time-series data cannot be stationary or non-stationary: it is only one realisation of the

process.

cov(yt,yt−k) is also referred to as an autocovariance.

 Autocovariances capture many salient properties of a stochastic process.

An autocorrelation is just the autocovariance scaled by the process variance, i.e. ρk,t =  .

  The scaling eliminates dependence on the unit of measurement (e.g. Vk,t  is higher for a process measured in cents than the same process measured in dollars; ρk,t  is the same).

  In general, ρk,t  ∈ (−1, 1).

For a stationary process, Vk  = V−k  and ρk  = ρ −k .

The autocorrelation function (ACF) is the plot of ρk  against k = 1, 2, . . . .

  The ACF describes all the autocorrelations in a stationary process.

Another way to measure the relationship between yt  and yt−k  is to compute the correlation with the influence of all intermediary yt−1 , . . . ,yt−k+1  “filtered out.”

This is known as the partial autocorrelation ϕkk .

For a stationary process, plotting ϕkk  against k generates the partial autocorrelation function (PACF).

Things to note:

  ϕ 11 = ρ 1 ;

  in general, the PACF is derived from the ACF.

One simple way to model a stochastic process is with the “regression”:

yt = a0 + a1yt−1 + ut .

This is called the first order auto-regressive model, or AR(1).

To make it useful in practice, we need assumptions about ut .

The “classical regression” assumptions are:

  Mean-independence:  E(ut|yt−1,yt−2 , . . . ) = 0.

  Homoscedasticity: Var(ut|yt−1,yt−2 , . . . ) = σu(2) .

Mean-independence is crucial, but homoscedasticity can be relaxed.

Mean-independence implies zero-autocorrelation: corr(ut,ut−k) = 0 for k = 1, 2, . . . .

What if the errors u1 , . . . ,uT  are also correlated?

Correlated errors could be modelled, for example, by

ut = et + b1et−1 ,

where et  is the uncorrelated innovation in the process.

The above is called a first-order moving average MA(1) process for ut .

In this case, assumptions are placed on et:

  Mean-independence:  E(et|yt−1,yt−2 , . . . ) = 0.

  Homoscedasticity: Var(et|yt−1,yt−2 , . . . ) = σe(2) .

Putting the AR(1) and and MA(1) together, we get:

yt = a0 + a1yt−1 + b1 εt−1 + εt ,

where {εt} satisfy mean-independence, zero-correlation and homoscedasticity. This is the autoregressive moving average model ARMA(1, 1).

In general, we work with a model containing p autoregressive lags and q moving average lags,

i.e. the ARMA(p,q):

p                                     q

工                     工

j=1                                j=1

Things to note:

  b1 = ··· = bq  = 0 implies an AR(p) process for yt;

  a1 = ··· = ap = 0 implies a MA(q) process for yt;

To analyse the properties of ARMA(p,q) models, it helps to define some notation.

Definition

The lag operator L applied to a stochastic process {yt} transforms a realisation at time t into a realisation at time t − 1, i.e.

yt−1 = Lyt .

This helps us write polynomials in the lag operator:

  a(L) = 1 − a1L − · · · − ap Lp ,

  b(L) = 1 + b1L + ··· + bq Lq .

Then, the ARMA(p,q) can be concisely expressed a(L)yt = a0 + b(L)εt .

Expected value of yt  conditional on yt−h,yt−h −1 , . . . :

E(yt|yt−h,yt−h −1 , . . . ) = E(a0 + a1yt−1 + εt|  ·  )

= a0 + a1E(yt−1 |  ·  ) + E(εt|  ·  )                  = a0 + a1 (a0 + a1E(yt−2 |  ·  ) + E(εt−1 |  ·  ))

= (1 + a1 + a1(2) + ··· + a1(h)−1) a0 + a1(h)yt−h = a0 + a1(h)yt−h .

The unconditional mean E(yt) is the limiting case as h −→ ∞:

E(yt) =  lim  E(yt|yt−h,yt−h −1 , . . . ).

Taking the limit yields:

  E(yt|yt−h,yt−h −1 , . . . ) −→  if |a1 | < 1;

  E(yt|yt−h,yt−h −1 , . . . ) −→ indeterminate form (i.e. does not exist) if |a1 | ≥ 1.

Hence, a finite E(yt) exists if and only if |a1 | < 1.

The AR(1) model with |a1 | ≥ 1 is called unstable.

Instability implies non-stationarity, but not the other way around.

Variance of yt  conditional on yt−h,yt−h −1 , . . . :

Var(yt|yt−h,yt−h −1 , . . . ) = Var ((1 + a1 + a1(2) + ··· + a1(h)−1) a0 + a1(h)yt−h

+et + a1et−1 + a1(2)et−2 + ··· + a1(h)−1et−h+1 |  ·  ) = Var(et|  ·  ) + a1(2)Var(et−1 |  ·  )

+ a1(4)Var(et−2 |  ·  ) + ··· + a1(2)(h− 1)Var(et−h+1 |  ·  )

= (1 + a1(2) + a1(4) + ··· + a(h1(2) − 1) ) σε(2) = σε(2) .

Covariance between yt  and yt−k  conditional on yt−h,yt−h −1 , . . . :

cov(yt,yt−k|yt−h,yt−h −1 , . . . ) = a1(k)σε(2) ,

The unconditional variance and covariances are obtained in the limit as h −→ ∞ .

If the AR(1) is unstable, then the unconditional variance and covariances do not exist. Otherwise:

 V0 = 1 2εa1(2) ;

 Vk  =  , k = 1, 2, . . . ;

 ρk  = a1(k) , k = 1, 2, . . . ;

  ϕ 11 = a1 , ϕkk  = 0 for all k ≥ 2.

The ACF of a stable AR(1) decays geometrically as k −→ ∞ .

The PACF of a stable AR(1) vanishes for all k ≥ 2.

The unconditional mean of yt  is:

E(yt) = E(a0 + b1et−1 + et) = a0 .

The unconditional variance of yt  is:

Var(yt) = Var(a0 + b1et−1 + et) = (1 + b1(2))σε(2) .

The unconditional covariance between yt  and yt−k  is:

cov(yt,yt−k) = cov(a0 + b1et−1 + et,a0 + b1et−k −1 + et−k)

= E((b1et−1 + et)(b1et−k −1 + et−k))

= b1 σε(2)  if k = 1; 0 for all k ≥ 2.

Moments conditional on yt−1,yt−2 , . . . , etc. are complicated, but yt  is independent of yt−2 ,yt−3 , . . . .

The MA(1) is always stable: the unconditional mean, variances and covariances always exist (since Var(εt) exists by assumption).

If |b1 | > 1, then the MA(1) is not invertible, but this does not affect the ACF (we will return to non-invertibility).

The ACF of an MA(1) always exists and is given by ρ 1 =  , ρk  = 0 for all k ≥ 2.

The PACF of an MA(1) always exists, with ϕ 11 =  and ϕkk  decaying geometrically as

k −→ ∞ .

  Recall that the PACF is computed form the ACF, so if the ACF exists, the PACF does as

well.