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What univariate models tell us about multivariatemacroeconomic models
James Mitchell (Warwick), Donald Robertson (Cambridge) andStephen Wright (Birkbeck)
NBP Workshop on Forecasting 2016 (Warsaw, Poland)
21 November 2016
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 1 / 25
Puzzle 1: The Predictive Puzzle
Multivariate macro models often fail to outpredict univariate models
Evidence from Nelson (1972, AER) through to Stock & Watson(2015)...
AR(MA) is a tough benchmark to beat
for inflation, exchange rates, stock prices, consumption...it’s challenging to find predictor variables that persistently predict
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 2 / 25
Puzzle 2: The Order Puzzle
The univariate models that compete with multivariate models inpredictive terms tend to be of low order
But the multivariate models commonly have large numbers of statevariables
which should imply high-order ARMA reduced forms (see Wallis, 1977Econometrica; Cubbada et al., 2009 Journal of Econometrics)
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 3 / 25
Our Aims
The paper offers an explanation for both Puzzles - in terms ofpopulation properties
we work backwards, observing just the univariate history of yt , datafor which are assumed to be generated by an (unknown) multivariatemacroeconomic model and show that
univariate properties can tightly constrain the properties of the (truebut unknown) multivariate macro model
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 4 / 25
Our key results - The predictive puzzle
1 We show how much better we could forecast yt if we conditioned onthe true state variables of the underlying multivariate model, ratherthan just its history, y t
The one-step-ahead predictive R2 must lie between bounds, R2min andR2max, that are strictly within [0, 1]
bounds derive from the ARMA parameters based on y t
results generalise to time-varying parameter models
the gap between R2min and R2max can be narrow for some yt processes
searching for reliable predictors is then a thankless task
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 5 / 25
Our key results - The order puzzle
2. If yt is ARMA(p, q) in population then (absent tight restrictions) the# unique predictors from the macro model, r = q
In finite samples, p and q may never be knowableSo to explain the order puzzleeither, the true multivariate model does have many state variables, butit’s close to satisfying parameter restrictions that imply (near)cancellation of AR and MA roots in the reduced form. Why?or, there are only a few distinct eigenvalues driving the macroeconomy- as in factor models
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 6 / 25
The true multivariate model
Let yt be generated by a multivariate macroeconomic model in then-vector of states zt
zt = Azt−1 +Bstyt = Czt−1 +Dst
Under (weak, and standard) Assumptions, ABCD implies a truepredictive regression for the first element of yt , where r ≤ n due to(possible) linear dependence between the states
yt = β′xt−1 + utxt(r×1)
= Mxt−1 + vt
where M =diag(µ1, ..., µr ) contains the distinct eigenvalues of A
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 7 / 25
Assumptions
A1 The autoregressive matrix A of the state variables zt can bediagonalised as A = T−1M∗T where M∗ is an n× n diagonal matrix, withfirst r diagonal elements M∗ii = µi , i = 1, ..., r , r ≤ n being the distincteigenvalues of AA2 |µi | < 1, i = 1, ..., rA3 st is an s × 1 vector of mutually orthogonal Gaussian IID processeswith E (sts′t ) = IsA4 DD′ has non-zero diagonal elements.
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 8 / 25
Lemma(The Macroeconomist’s ARMA) The true predictive regression and theprocess for the associated true predictor vector xt−1 together imply that ythas a unique fundamental ARMA(r , r) representation with parametersλ = (λ1, ...,λr ) and θ = (θ1, ..., θr )
λ (L) yt = θ (L) εt
where λ (L) ≡ ∏ri=1 (1− λiL) ≡ det (I −ML) and θ (L) ≡
∏ri=1 (1− θiL), |θi | ≤ 1, ∀i
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 9 / 25
The macroeconomist’s ARMA - a simple illustration
Consider an ABCD model with a single state variable zt and a 2× 1vector of structural shocks. This implies the predictive system
yt = βxt−1 + utxt = µxt−1 + vt
with A = µ, C = β, vt = Bst , ut = DstThis system is common in predictive return regressions
with yt some measure of returns or excess returns and xt somestationary valuation criterion
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 10 / 25
The macroeconomist’s ARMA(1,1) - a simple illustration
By substitution
(1− µL) yt = βvt−1 + (1− µL) ut
i.e. yt admits a fundamental ARMA(1,1) representation
(1− µL) yt = (1− ψL) εt
with |ψ| < 1, where ψ = ψ (A,B,C,D) is the solution to
−ψ
1+ ψ2=
µσ2u − βσuv
(1+ µ2) σ2u + β2σ2v + 2µβσuv
ψ−1 is also a solution
but this results in a nonfundamental representation, with a differentIID shock process, which cannot be recovered from the history y t (werevert to this representation in due course)
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 11 / 25
What the ARMA tells us about the true predictiveregression
We now work backwards, and ask: if we only observed the history ofyt , what would its univariate properties (as captured by λ and θ) tellus about the underlying multivariate model?
They do not, in general, tell us about the structural shocks stst ∝ εt only when n = 1 and the ARMA is fundamental
But the ARMA is informative about some properties of theunderlying multivariate model
1 The predictive power of the multivariate model, expressed in terms ofits one-step-ahead predictive R2, must lie between bounds definedw.r.t. the ARMA parameters
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 12 / 25
Bounds for the predictive R-sq
Under A1-A4, the predictive R2 (A,B,C,D) = 1− σ2u/σ2y of theregression derived from the underlying ABCD macro model
0 < R2min ≤ R2 ≤ R2max < 1
where R2min is the predictive R2 from the ARMA representation and so is a
function of λ and θ alone, and
R2max = R2min +
(1− R2min
)(1−
r
∏i=1
θ2i )
so R2max is also a function of λ and θ alone
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 13 / 25
The fundamental ARMA(1,1)
(1− λL) yt = (1− θL) εt
λ and θ are less than unity in modulus
εt are then derived from the history of yt
εt =
(1− λL1− θL
)yt =
∞
∑i=0
θi (yt−i − λyt−1−i )
andR2min = R
2F = 1− σ2ε /σ2y
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 14 / 25
The non-fundamental ARMA(1,1)
yt =
(1− θ−1L1− λL
)ηt
Now the (structural) shocks cannnot be derived from the history of yt .Instead
ηt =
(1− λL
1− θ−1L
)yt = −θF
(1− λL1− θF
)yt
= −∞
∑i=1
θi [yt+i − λyt+i−1]
where F = L−1
ηt is a linear combination of current and future values of yt
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 15 / 25
Bounds for the predictive R-sq: ARMA(1,1)
R2min = R2F =(θ − λ)2
1− λ2 + (θ − λ)2
Rmax = R2min +(1− R2min
) (1− θ2
)=
(1− λθ)2
1− λ2 + (θ − λ)2
If θ is close to λ, yt is close to white-noise, and R2min is close to 0
If θ is close to 0, R2max is close to 1
Iff θ and λ are both suffi ciently close to 0 (yt and xt are white-noise)the bound opens up to [0, 1]
As |θ| → 1, both R2min and R2max →
1−sgn(θ)λ2 : i.e. the R2 bound
collapses to a single point
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 16 / 25
Intuition
Lower bound: the predictions from the fundamental ARMA conditiononly on the history of yt
they cannot be worsened by increasing the information set to includethe true predictor
Upper bound: ηt has minimum innovation variance
ηt are a linear combination of the history and the future of yt
While we can improve our prediction of yt by using the true statevariable, there is a limit to this improvement
and this limit is determined by the univariate properties of yt
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 17 / 25
What the ARMA tells us about the dimensions of the truemultivariate model
Proposition(ARMA order and the ABCD Model) Let yt admit a minimalARMA(p, q) representation in population, with
q = r −#{θi = 0} −#{θi = λi 6= 0}p = r −#{λi = 0} −#{θi = λi 6= 0}
Under A1 to A3, the data yt must have been generated by a multivariatemodel in which, in the absence of exact restrictions over the parameters(A,B,C,D), A has r = q distinct eigenvalues. Thus there must be a truepredictive system for yt with r = q predictors
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 18 / 25
What the ARMA tells us about the dimensions of the truemultivariate model
CorollaryThe predictor vector xt in the true predictive regression can only haver > q elements if the parameters of the structural model (A,B,C,D)satisfy r − q exact restrictions that each ensure θi (A,B,C,D) = 0 forsome i, or θj (A,B,C,D) = λk (A,B,C,D), for some j and k .
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 19 / 25
Stock & Watson’s (2007) model of US inflation
Find little evidence that it’s possible to out-forecast a univariatemodel, particularly in recent data
Their preferred model lets the variances of the permanent andtransitory innovations vary over time: a UC-SV model
Their UC-SV is equivalent to a time-varying IMA(1,1) model ofinflation, yt = ∆πt
yt = (1− θtL) εt
where λ = 0
θt thus determines time-varying R2 predictive bounds
λ = 0 tells us that any good predictor (in composite) must be close toIID
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 20 / 25
Time-varying MA(1) parameter - median, 16.5% and83.5% quantiles of the posterior distribution
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 21 / 25
Time-varying R-sq bounds: posterior medians
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 22 / 25
Time-varying 16.5% and 83.5% quantiles for R-sq max -R-sq min
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 23 / 25
Implications of Stock & Watson’s (2007) model of inflation
The changing univariate properties of inflation noted by SW dictatethat even the true multivariate model would now struggle toout-predict a univariate model
The IMA(1,1) model implies that, in the absence of cancellation,there is a single IID predictor
No wonder macroeconomists struggle to find output gap measures thatforecast inflationMore promise in looking for “news” type predictorsWhile higher-dimensional multivariate models (with cancellation) canescape the R2 bounds, why should this be a feature of themacroeconomy? Smets and Wouters’(2007) DSGE model does notescape the bounds, as its predictions for inflation are too persistent
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 24 / 25
Conclusions
What we know about the univariate time-series properties of yt tellsus a lot about the properties and performance of the true (butunknown) multivariate predictive model
Helps us explain why it’s often hard to out-forecast a univariate model
And why researchers often struggle to find reliable predictors; e.g. whythe output gap struggles to forecast inflationMultivariate models with news variables and/or with few distincteigenvalues may fare better?
Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 25 / 25