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(School of Economics) Applied Econometrics for Macro and Finance Week 2 1 / 28
Forecasting Univariate Processes – I
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Forecasting in Practice
Predictions about future events are inherent to all decisions.
In policy, business and finance decisions, uncertainty related to future outcomes represents
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.
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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.
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Reserve Bank of Australia Economic Outlook Report, August 2021
Forecast scenarios, December 2019 = 100
Sources: ABS; RBA
(School of Economics) Applied Econometrics for Macro and Finance
Institute for Health Metrics and Evaluation COVID-19 Projections
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Matching and Extrapolating Patterns
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(School of Economics)
Applied Econometrics for Macro and Finance
Matching and Extrapolating Patterns
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Matching and Extrapolating Patterns
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Matching and Extrapolating Patterns
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Matching and Extrapolating Patterns
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Matching and Extrapolating Patterns
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Matching and Extrapolating Patterns
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Stochastic Process
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}.
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Definition
A stochastic process is a collection of random variables that are ordered in time.
Realisations, Moments
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: γ0,t ≡ Var(yt) = E((yt − μt)2), t = 1, . . . , T ; i.e., the average of square deviations from the mean.
Covariance:γk,t ≡Cov(yt,yt−k)=E((yt −μt)(yt−k −μt−k)), t=k+1,…,T.
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Definition
A realisation is one of a (typically) infinite set of values of yt, t = 1, . . . , T , randomly generated according to the probability distribution underlying the stochastic process.
Stationarity
Several forms of stationarity that can be used to describe a stochastic process. We keep things simple with the following.
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.
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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) = σy2 = γ0,
Cov(yt, yt−k) = γk, k ≥ 1,
Autocorrelation Functions
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 = γk,t . γ0,t
The scaling eliminates dependence on the unit of measurement (e.g. γk,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, γk = γ−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.
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Partial Autocorrelation Function (PACF)
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
Things to note: φ11 = ρ1;
in general, the PACF is derived from the ACF.
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The First-Order Autoregressive Model
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, . . . ) = σu2.
Mean-independence is crucial, but homoscedasticity can be relaxed.
Mean-independence implies zero-autocorrelation: corr(ut, ut−k) = 0 for k = 1, 2, . . . .
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The First-Order Moving Average Model
Whatiftheerrorsu1,…,uT arealsocorrelated? Correlated errors could be modelled, for example, by
ut = εt + b1εt−1, where εt 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 εt: Mean-independence: E(εt | yt−1, yt−2, . . . ) = 0. Homoscedasticity: Var(εt | yt−1, yt−2, . . . ) = σε2.
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The Autoregressive Moving Average Model
Putting the AR(1) and and MA(1) together, we get:
yt =a0 +a1yt−1 +b1εt−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):
yt = a0 + aj yt−j + εt + bj εt−j .
Things to note:
b1 = ··· = bq = 0 implies an AR(p) process for yt; a1 = ··· = ap = 0 implies a MA(q) process for yt;
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Lag Operator Notation
To analyse the properties of ARMA(p, q) models, it helps to define some notation.
This helps us write polynomials in the lag operator: a(L) = 1 − a1L − · · · − apLp,
b(L) = 1 + b1L + · · · + bqLq.
Then, the ARMA(p, q) can be concisely expressed a(L)yt = a0 + b(L)εt.
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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.
Moments of the AR(1) Process
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 +a21 +···+ah−1a0 +ah1yt−h 1
= 1 − a h1 a 0 + a h1 y t − h . 1−a1
(School of Economics)
Applied Econometrics for Macro and Finance
Moments of the AR(1) Process
The unconditional mean E(yt) is the limiting case as h −→ ∞:
E(yt)= lim E(yt|yt−h,yt−h−1,…). h→∞
Taking the limit yields:
E(yt |yt−h,yt−h−1,…) −→ a0 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.
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Moments of the AR(1) Process
Variance of yt conditional on yt−h, yt−h−1, . . . : Var(yt|yt−h,yt−h−1,...)=Var1+a1 +a21 +···+ah−1a0 +ah1yt−h
+εt+a1εt−1+a21εt−2+···+ah−1εt−h+1| · 1
=Var(εt| · )+a21Var(εt−1| · )
+a4Var(ε | · )+···+a2(h−1)Var(ε
= 1+a21+a41+···+a1 σε2=
Covariance between yt and yt−k conditional on yt−h, yt−h−1, . . . : 1 − a2(h−k)
cov(yt,yt−k |yt−h,yt−h−1,...) = 1 ak1σε2, 1 − a 21
1 t−2 1 t−h+1
(School of Economics) Applied Econometrics for Macro and Finance
The ACF and PACF of an AR(1) Process
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:
γ = σε2 ; 0 1−a21
γ =ak1σε2,k=1,2,...; k 1−a21
ρk = ak1, k = 1,2,...;
φ11 =a1,φkk =0forallk≥2.
The ACF of a stable AR(1) decays geometrically as k −→ ∞. The PACF of a stable AR(1) vanishes for all k ≥ 2.
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Moments of the MA(1) Process
The unconditional mean of yt is:
E(yt) = E(a0 + b1εt−1 + εt) = a0.
The unconditional variance of yt is:
Var(yt) = Var(a0 + b1εt−1 + εt) = (1 + b21)σε2.
The unconditional covariance between yt and yt−k is:
cov(yt, yt−k) = cov(a0 + b1εt−1 + εt, a0 + b1εt−k−1 + εt−k) = E ((b1εt−1 + εt)(b1εt−k−1 + εt−k))
=b1σε2 ifk=1;0forallk≥2.
Moments conditional on yt−1, yt−2, . . . , etc. are complicated, but yt is independent of yt−2,yt−3,....
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The ACF and PACF of an MA(1) Process
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).
TheACFofanMA(1)alwaysexistsandisgivenbyρ1 = b12,ρk =0forallk≥2.
The PACF of an MA(1) always exists, with φ11 = b1 2 and φkk decaying geometrically as
k −→ ∞. well.
Recall that the PACF is computed form the ACF, so if the ACF exists, the PACF does as
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The ACF and PACF of AR(p), MA(q) and ARMA(p, q) Processes
In general, we can summarise ACFs of PACFs of ARMA processes as follows.
For a pure AR(p), the ACF and PACF exist if only if it is stable, in which case the ACF decays to zero as k −→ ∞;
the PACF is given by
φ11,…,φpp computedfromtheACF,with φ11 =ρ1,φpp =ap and
φkk =0forallk≥p+1.
For a pure MA(q), the ACF and PACF always exist and
the ACF vanishes for all k ≥ q + 1;
the PACF is computed from the ACF, with φ11 = ρ1 and φkk decaying as k −→ ∞.
For a general ARMA(p, q), the ACF and PACF exist if and only if it is stable, in which case
both decay as k −→ ∞.
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