To evaluate the forecasting ability of the IPSO, we use pairwise vector autoregressive models (VARs). In each VAR, we use the IPSO and one other macroeconomic time series.
For each pair, we conduct a Granger causality test, i.e., we test the hypothesis whether past values of the IPSO can help to predict inflation or consumption expenditure. The analysis is conducted in R using the vars-package (Pfaff 2008). In a first analysis, we only consider a VAR(1). We then extend the analysis on the basis of the Schwarz-Bayes Information Criterion (BIC) to select the lag length of the model p.
9 Board of Governors of the Federal Reserve System (US), 5-Year Treasury Constant Maturity Rate, re-trieved fromhttps://www.federalreserve.gov/datadownload/Choose.aspx?rel=H15, Unique Iden-tifier: H15/H15/RIFLGFCY05 XII N.B
10 Board of Governors of the Federal Reserve System (US), 5-Year Treasury Constant Maturity Rate, re-trieved fromhttps://www.federalreserve.gov/datadownload/Choose.aspx?rel=H15, Unique Iden-tifier: H15/H15/RIFLGFCY05 N.B
Figure 1.11: IPSO Growth and Macroeconomic Time Series
The figure shows in each panel the growth of the IPSO for the Euro Area or the US constructed on search-terms for the price levels P = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 20; 30; 40; 50; 60; 70; 80; 90; 100; 200;
300; 400; 500; 600; 1,000} in red. The black lines are either US consumption growth, US inflation, EA inflation or EA consumption loan growth.
2008 2010 2012 2014 2016 2018
−1.5−1.0−0.50.00.51.0
Time
Inflation −30−20−100102030 IPSO Growth
(a)∆EUR-IPSO and EA-Inflation
2008 2010 2012 2014 2016 2018
−2.0−1.00.00.51.0
Time
Inflation −20−10010 IPSO Growth
(b) ∆USD-IPSO and US-Inflation
2008 2010 2012 2014 2016 2018
−40−200102030
Growth of Loans to Households For Consumption Time −30−20−100102030 IPSO Growth
(c) ∆EUR-IPSO and
EA Consumption Loan Growth
2008 2010 2012 2014 2016 2018
−1.0−0.50.00.5
Time
Consumption Growth −20−10010 IPSO Growth
(d)∆USD-IPSO and
Growth of US-Consumption
The model set up for the VAR(p) is xi,t=µi+
pi
∑
j=1
Ai,jxi,t−j+
S−1
∑
r=1
bi st−r+εi,t, (1.7) wherexi,t is a 2×1 vector that contains the logarithmic growth rate (in percent) of the IPSOt as well as the logarithmic growth rate (in percent) of the variable of interest in the various specifications. For monthly data, we only consider logarithmic growth rates as the level prices are non-stationary and we are interested whether or not changes in the IPSO are helpful to predict consumption growth or inflation. Also in the case of the daily BEIR, we consider logarithmic growth rates.
The subscript i refers to the different combinations of time series. Ai,j are the 2×2 parameter matrices while µi is a vector of constants. st is a centered seasonal control variable. When controlling for annual seasonality, i.e., S=12, we include S−1=11 lags of
the control variable st. It is constructed as st=Dt−S1 ∑12r=1Dt whereDtis a month specific dummy variable which is 1 for a certain month and otherwise 0. In all of our VAR-models for monthly time series, we control for annual seasonality. In the case of the BEIR time series, on a daily frequency, we do not include any seasonal control variables. We assume the innovationsεi,t to be i.i.d. pi is the lag-length, either fixed to 1 or selected by the BIC.
To test for Granger causality (Granger 1969), we use the heteroskedasticity consistent jackknife estimator ofEfron (1982). Furthermore, we test for contemporaneous correlation, termed instantaneous Granger causality, between the measures based on the test-statistic developed by Granger (1969). However, Granger causality tests the in-sample relevance of the measure. In order to assess the out-of-sample forecasting ability, we analyze the out-of-sample forecasting performance with a rolling one-step ahead forecast. For each prediction xi,t+1, we re-estimate a model based on the preceding L = 84 monthly observations equivalent to the last 7 years of monthly data. For the break-even inflation rate, which can be measured on a daily frequency, we use the last L=250 observations equivalent to the observations for the business days of the last year. In each window the lag-length is either fixed to 1 or newly selected by the BIC.
To evaluate the forecasts, we report the out-of-sample root mean squared prediction error RM SP Ei of the predicted macroeconomic time series,
RM SP Ei=
¿ ÁÁ
À 1
T −L
T−1
∑
t=L
(xi,t+1−xˆi,t+1)2,
wherexi,t+1is the observed macroeconomic variable of interest, i.e., inflation or consumption.
For simplicity, we mostly report theRM SP Ei only for out-of-sample forecasts in which the lag-length in each rolling window has been selected by the BIC. However, in the case of the test developed by Clark and West(2006,2007), we have to use a one-size-fits-all estimation strategy in which we fix the lag-length to pi = 1 across all forecasting windows. Only then, we can test whether the RM SP E is significantly reduced when including the IPSO by using the methodology of Clark and West (2006, 2007) for nested models. For that purpose, we have to estimate a baseline model without the IPSO on the macroeconomic time series of interest. The base model needs to be nested in the extended model including the IPSO. In our case the base model is an auto-regressive model of order one, AR(1) and the extended model is a VAR(1), including the IPSO. The null-hypothesis of the test is that including the IPSO in the model setup does not affect the out-of-sample RM SP Ei and will result in the same forecast error as for the base model.
The test statistic is calculated as
z=RM SP E0− (RM SP Ei−κi),
whereRM SP E0is the root-mean-squared error of the base model,κi =T1−i∑Tt=pxˆi,t+1−ˆx0,i,t+1 and ˆx0,i,t+1 denotes the forecast of the base model for the macroeconomic time series i.
Critical values are available from Clark and West (2006,2007).
We also calculate the R2 of the Mincer and Zarnowitz (1969) (R2M Z) regression of the realizations xi,t+1 on the out-of-sample forecasts ˆxi,t+1 from the respective model. The regression equation reads
xi,t+1=a0+a1xˆi,t+1+em,t+1. The higherR2M Z the better the forecast.