SIMPLE ROBUST TESTS FOR THE SPECIFICATION OF HIGH-FREQUENCY PREDICTORS OF A LOW-FREQUENCY SERIES

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1 SIMPLE ROBUST TESTS FOR THE SPECIFICATION OF HIGH-FREQUENCY PREDICTORS OF A LOW-FREQUENCY SERIES J. Isaac Miller University of Missouri International Symposium on Forecasting Riverside, California June 23, 2015

2 INTRODUCTION Research Problem Forecast or nowcast a low-frequency (LF) target High-frequency (HF) predictor available Research Questions Should we aggregate the HF data? How? Should we consider a mixed-frequency (MF) specification? Solutions Variable addition test (VAT) statistics (Wu, 1973) Approach unifies and nontrivially extends that of Andreou, Ghysels, and Kourtellos (2010) AGK10... Miller (2014) M14

3 FORECASTING PROBLEM Simple Forecasting Problem Forecast a LF series (y t ), t = 1,...,T Use a HF predictor (x t i/m ), i = 1,...,m < Aggregation Notation Let xt 1:m = (x t,x t 1/m,...,x t (m 1)/m ), m HF observations Let π = (π 1,..., π m ), aggregation weights Let xt m = xt 1:m π, single aggregated LF predictor Question: How can we choose or estimate π? Choose = aggregate predictor Estimate = explicitly use HF information from the predictor Can we do so in a way that is robust to order of integration (unit roots) and biased forecasts?

4 THREE AGGREGATION WEIGHT SPECIFICATIONS Aggregated Predictor, π Fully Specified y t+1 = x m t β+ε t+1 β m = 0 minimizes mean squared forecast error (MSFE) Use ˆβ : Scalar least squares estimator Disaggregated Predictor, No Restrictions y t+1 = x 1:m t α+ε t+1 α = πβ estimated y t+1 = α 1 x t + α 2 x t 1/m + +α p x t (m 1)/m + ε t+1 (mixed frequency distributed lag model) Use ˆα : Vector least squares estimator Disaggregated Predictor, Partially Specified y t+1 = x 1:m t α(θ)+ε t+1 Parsimonious, nonlinear, parametric specification α(θ), called MIxed DAta Sampling (MIDAS) (Ghysels et al., 2004) Use α( ˆθ) : Vector nonlinear least squares estimator

5 THREE SETS OF HYPOTHESES Set τ 1 (Focus of this presentation) H 0 : Fully specified, LF predictors H A : Fully unrestricted, HF predictors (DL) Set τ 2 H 0 : Fully specified, LF predictors H A : Partially specified, HF predictors (MIDAS) Set τ 3 H 0 : Partially specified, HF predictors (MIDAS) H A : Fully unrestricted, HF predictors (DL)

6 VAT TESTING STRATEGY INTUITION Problem: m may be large relative to T Conventional tests (e.g., Kvedaras and Zemlys, 2012, for τ 3 ) may be infeasible/impractical against DL alternative (τ 1 or τ 3 ) Solution: VAT statistic (AGK10 for τ 2, M14 for τ 3 ) Intuition Choose q m elements of xt 1:m (AGK10)... linearly independent combinations of xt 1:m (M14) Under H 0 : xt m = xt 1:m π correctly specified Elements/combinations of xt 1:m are superfluous Under H A : (α πˆβ) = O p (1) = omitted variable xt 1:m (α πˆβ) Elements/combinations of xt 1:m detect omitted variable

7 VAT TESTING STRATEGY IMPLEMENTATION Step 1: Step 2: Null model: y t+1 = x m t β+ε t+1 Estimate null model, retain fitted residuals (ˆε t+1 ) Test regression: ˆε t+1 = x 1:m t Υϕ+e t+1 Υ m (q+1) includes π ˆβ from the null model... q additional linearly independent combinations of x 1:m t Test H 0 : ϕ q = 0 against H A : ϕ q = 0 (last q elements of ϕ) Wald statistic: V T = T ˆϕ q ˆΩ 1 q ˆϕ q ˆϕ q : least squares estimator of ϕ q ˆΩ q : consistent estimator of asymptotic variance of ˆϕ q (robust to heteroskedasticity, serial correlation)

8 ASYMPTOTIC PROPERTIES WITH I(0) SERIES Theorem Consider the test statistic V T with I(0) series (x t i/m ). Remarks Under H 0, V T d χ 2 q, while Under H A, V T = O p (T). H 0 : χ 2 q asymptotic distribution, standard critical values Exploits asymptotic normality using I(0) series Robust to serial correlation, heteroskedasticity, biased forecasts AGK10 showed for τ 2 under very specific assumptions M14 showed for τ 3 H A : Consistent against alternative

9 ASYMPTOTIC PROPERTIES WITH I(1) SERIES Theorem Consider the test statistic V T with I(1) series (x t i/m ). Remarks Under H 0, V T = O p (1), but Under H A, V T = O p (T). H 0 : Unexpected, messy limiting distribution Size distortion if χ 2 q critical values used for τ 1,τ 2 M14 showed standard χ 2 q critical values for τ 3 H A : Consistent against alternative

10 A MODIFIED VAT STATISTIC Intuition Add an iid disturbance with slowly increasing variance Sacrifice power to correct size distortion Implementation Define ˆε t+1 = ˆε t+1+h(t)u t+1 u t iidn(0,1) h(t) while h(t)t 1/2 0 as T Rule of thumb: h(t) = σ u T ǫ 0 < ǫ 1/2 σ u = RMSE(ˆε t+1 ) (work in progress) Modified test regression: ˆε t+1 = x1:m t Υϕ+e t+1 Modified VAT statistic: V T = qf (F is an F-statistic of H 0 : ϕ q = 0)

11 ASYMPTOTIC PROPERTIES WITH I(0) OR I(1) SERIES Theorem Consider the test statistic V T with I(0) or I(1) series (x t i/m). Remarks Under H 0 of all tests, V T d χ 2 q, while Under H A of all tests, V T = O p(t). H 0 : χ 2 q asymptotic distribution, standard critical values H A : Consistent against alternative Power loss against local alternatives in small samples

12 SMALL-SAMPLE SETUP High-Frequency Data Generating Process y t+1 i/m = βx t i/m + ε [ ] [ t+1 i/m εt+1 i/m 0 0 = x t i/m 0 d ( [ ]) 1 ρ ξ t i/m iidn 0, ρ 1 d = 0,1 : order of integration of (x t im ) ρ = 1/2, β = 10, 1,000 simulations ][ εt+1 (i+1)/m x t (i+1)/m Aggregate (y t+1 i/m ) = Mixed Frequencies Span fixed: T = 200 Frequency varied: m = {4,150,365} Test Implementation ǫ = {0,1/5,1/2}, σ 2 u = 1 Results show qf ] + ξ t+1 i/m

13 POWER FUNCTIONS Power function π j,2 (θ) = (2 j/m) 4θ / m u=1(2 u/m) 4θ for θ = [0,2] H 0 : θ = 0, flat sampling H A : θ = 0 (θ = 2 is end-of-period, EOP, sampling) Unrestricted (AGK10) VAT Statistic (q = m 1) First column of Υ: π 0 Remaining: last m 1 columns of m m identity matrix Parsimonious (M14) VAT Statistic (q = 2) First column of Υ: π 0 Second column: 0.9 j 1 / m u=10.9 u 1 Third column: 2(m+1 j)/(m(m+1))

14 POWER FUNCTIONS Note: Nominal size is 5%.

15 PRACTICAL CONSIDERATIONS Some Recommendations Choose small q. Otherwise, (worse) size distortion, power loss, test infeasibility Use V T with ǫ = 1/5 for τ 1 (and τ 2 ), unless I(1) series ruled out Sequential Procedure 1 Use VT to test for nulls of interest, such as flat sampling or end-of-period sampling. Fail to Reject: Use that sampling scheme Reject: Use a parameterized mixed-frequency structure 2 Use V T to test parameterized MF structure against a more general alternative (τ 3 ) Fail to Reject: Use that MF parameterization Reject: Use unrestricted distributed lag (if feasible) or repeat step 2 with a different MF parameterization

16 MONTHLY STATE RETAIL GASOLINE PRICES Why State Gasoline Prices? Much variation in taxes and fees across states Pennsylvania: 50.50/c per gallon (#1) California: 45.39/c per gallon (#2) Michigan: 30.26/c per gallon (#17) Missouri: 17.30/c per gallon (#47) Alaska: 11.30/c per gallon (#51) Forecast or conditional forecast may be important to state policy makers Nowcast may be important to consumers, especially near state borders

17 SOME RETAIL GASOLINE DATA SOURCES Public Sources US DOE, Energy Info Admin (EIA) < Monthly, stopped in 2011 for all but nine states MO Dept of Econ Development, Div of Energy < 1-2 times monthly, pdf with little historical data U of MO, Econ & Policy Analysis Research Center < Monthly, pdf with current and historical statistics (y t ) Private Sources ($$$) AAA Daily Fuel Gauge Report < GasBuddy.com <

18 SOME USEFUL PREDICTORS Useful Predictors of Monthly Retail State Prices Regional aggregate Weekly Midwest (PADD2) retail prices from EIA Aggregate from many states (and state taxes and fees) Differences in state taxes/fees stable across time Calendar effect: month t contains 4-5 weeks m t Monthly series: (x 1:m t 1t ) Gulf Coast Conventional Gasoline Regular Spot Price FOB Daily, market-traded spot price from EIA Free on board: no taxes, transportation costs, etc. Taxes, etc. stable over time especially within a month Calendar effect: month t contains business days n t Monthly series: (x 1:n t 2t )

19 FORECASTING AND NOWCASTING MODELS Forecasting Model: y t+1 = α 0 +x 1:m t 1t α 1t +x 1:n t 2t α 2t + ε t+1 HF predictors: all current information relevant to future y t+1 Dimensions of α 1t, α 2t vary due to calendar effects Nowcasting Model: y t = α 0 +x 1:m t 1,t α 1,t +x 1:n t 2,t α 2,t + ε t Relies only on contemporaneous information

20 DIFFERENT FREQUENCIES & CALENDAR EFFECTS HOW TO HANDLE? Option 1: Aggregate x m t 1t = x 1:m t 1t π mt and x n t 2t = x1:n t 2t π nt, where π mt, π nt take average of weeks/days in a month (flat) 2... last week/day of each month (EOP) 3... first week/day of each month (BOP) Option 2: Mixed-frequency distributed lag model Let m = min(m t ) = 4 and n = min(n t ) = 18 Beginning-weighted : x1t 1:m α 1 +x2t 1:n α 2 + ε bw t+1 Uses only the first 4 weeks and first 18 days End-weighted : x (m t m+1):m t 1t α 1 +x (n t n+1):n t 2t α 2 + ε ew t+1 Uses only the last 4 weeks and last 18 days

21 DIFFERENT FREQUENCIES & CALENDAR EFFECTS HOW TO HANDLE? Option 3: MIDAS with exponential Almon lag Beginning-aligned : π i,almon (γ 1 ) = exp(γ 1 i)/ m t u=1 exp(γ 1u) π j,almon (γ 2 ) = exp(γ 2 j)/ n t u=1 exp(γ 2u) for i = 1,...,m t, j = 1,...,n t. End-aligned : Let m = max(m t ) = 5 and n = max(n t ) = 23 π i,almon (γ 1 ) = exp(γ 1 i)/ m u=m m t +1 exp(γ 1u) π j,almon (γ 2 ) = exp(γ 2 j)/ n u=n n t +1 exp(γ 2u) for i = m m t +1,...,m, j = n n t +1,...,n.

22 FORECASTING TEST RESULTS FORECASTING MODEL Forecasting RMSE V T VT Flat-sampled EOP-sampled BOP-sampled End-aligned MIDAS Beginning-aligned MIDAS End-weighted DL Beginning-weighted DL Notes: σ u = RMSE(ˆε t+1 ), ǫ = 1/5, size-5% χ 2 4 crit. value:

23 MIDAS FORECASTING WEIGHTS FORECASTING MODEL MIDAS(5) Beginning-Aligned MIDAS(4) Week 5 Week 4 Week 3 Week 2 Week 1 Day 23 MIDAS(23) Beginning-Aligned MIDAS(22) Beginning-Aligned MIDAS(21) Beginning-Aligned MIDAS(20) Beginning-Aligned MIDAS(19) Beginning-Aligned MIDAS(18) Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day

24 NOWCASTING TEST RESULTS NOWCASTING MODEL Nowcasting RMSE V T VT Flat-sampled EOP-sampled BOP-sampled End-aligned MIDAS Beginning-aligned MIDAS End-weighted DL Beginning-weighted DL Notes: σ u = RMSE(ˆε t+1 ), ǫ = 1/5, size-5% χ 2 4 crit. value:

25 MIDAS NOWCASTING WEIGHTS NOWCASTING MODEL Day 23 MIDAS(5) Beginning-Aligned MIDAS(4) End-Aligned MIDAS(4) Week 5 Week 4 Week 3 Week 2 Week 1 MIDAS(23) Beginning-Aligned MIDAS(18) End-Aligned MIDAS(22) End-Aligned MIDAS(21) End-Aligned MIDAS(20) End-Aligned MIDAS(19) End-Aligned MIDAS(18) Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day Day

26 SOME CONCLUSIONS Variable addition test statistics Simple way to test mixed-frequency forecasting specifications Various nulls may be tested Tests may be constructed to be quite robust (heteroskedasticity, serial correlation, biased forecasts) Limiting distributions are simple i.e., χ 2 in most cases Care must be taken for the null of fully restricted aggregation when I(1) series are suspected A simple modification of the VAT Becomes robust to this case Sacrifices power to correct size