predicting food prices using data from consumer surveys...
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Predicting Food Prices using Data from Consumer Surveys and Search
Jisung Jo and Jayson L. Lusk
Contact information:
Jisung Jo
Department of Agricultural Economics
Oklahoma State University
Stillwater, OK 74078
Jayson L. Lusk
Department of Agricultural Economics
Oklahoma State University
Stillwater, OK 74078
Selected paper prepared for presentation at the Southern Agricultural Economics Association
(SAEA) Annual Meeting, San Antonio, Texas, February 6-9, 2016.
Copyright 2016 by Jisung Jo and Jayson L. Lusk. All rights reserved. Readers may make
verbatim copies of this document for non‐commercial purposes by any means, provided that
this copyright notice appears on all such copies.
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Abstract: Predicting future food prices is important not only for projecting and adjusting the
cost of government programs but also for business and household planning. This study asks
whether unconventional consumer-oriented measures might be useful in the predicting
Bureau of Labor Statistics (BLS) Food and Beverages Consumer Price Index (CPI). We
investigate the ability of Internet search-based index related to food prices (the Google trends
index) and survey-based sentiment indices (the index of consumer sentiment) to predict
changes in food-related BLS prices from January 2004 to July 2015. We consider several
forecasting models and find that a vector autoregression model (VAR) results in the lowest
root mean square error and mean absolute percentage error. We also ask whether our model
can out predict USDA Economic Research Service food-related CPI forecasts. Rolling
window comparison and encompassing tests are conducted, and we find that our new model
including consumer-oriented measures outperforms the USDA model in terms of predictive
accuracy.
Keywords: consumer sentiment, Internet search, food prices, forecasting
JEL Codes: C53, Q11
.
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Although food represents a relatively small share of consumers’ budgets, changes in in food
prices can have an important impact on household well-being, particularly for lower income
consumers who spend a larger portion of the income on food than higher income consumers.
In fact, many economic analysts focus only on the “core” consumer price index (CPI), which
excludes food and energy prices, because of a belief that prices for these items are “volatile
and are subject to price shocks that cannot be damped through monetary policy” (Greenlees
and McClelland, 2008, p. 11). Coupling food price volatility with the fact that food is
frequently purchased implies that consumer may be more aware or attentive to changes in the
price food than other items, and in fact the data suggest poorer households ted to pay less for
the same food items than the rich, perhaps because of greater price sensitivity and search
behavior (Broda et al., 2009). As such, data related to consumers’ price expectations may be
useful in forecasting changes in the price of food.
Projecting food prices is of interest to participants of the food supply chain as well as
government agencies. Firms make production decision on price expectations, and
agribusiness firms hedge commodity and output prices based on expected prices. Moreover,
changing food prices have implications for a number of government programs such as the
supplemental nutritional assistance (SNAP) program, the woman, infants, and children (WIC)
program, and the school lunch program, among others. Because of the desire to anticipate
future food prices, there are a number of ongoing efforts to forecast the food component of
the CPI (e.g., Kuhns et al., 2015).
Virtually all of the existing efforts to forecast the food-related CPI utilize time series
models where future price changes are estimated as a function of past food prices, and lagged
values of related variables (Joutz, 1997). These models are thus backward looking. However,
there are a number of more forward-looking variables that might have utility in predicting
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food price changes. In this paper, we consider two such measures: a survey-based index (the
Index of Consumer Sentiment (ICS) from the University of Michigan) and a search-based
index, (Google Trends Index (GTI)).
Previous research suggests the potential for survey-based sentiment indices like ICS
to forecast future food prices even though it is an overall sentiment not just focused on food.
Wilcox (2007) found that ICS improves forecasts of consumption and expenditures on
durable as well as non-durable goods and service. Ang, Bekaert, and Wei (2006) also found
that survey forecasts outperform time series forecasts, an economic model of the Philips
curve, and information embedded in asset prices using the forecast comparison regression
followed by Stock and Watson (1999). Girardi et al (forthcoming) also found survey data as
useful in economic growth; they highlight the utility of using survey-data for “nowcasting”
given that releases of public data, such as the CPI, often occur with a significant lag.
In addition to survey-based measures, newer measures related to consumers’ internet
search behavior are now available. Internet users now represent 84.2% of United States of
America’s population (World Bank, 2013). Prior research has showed some promise in using
measures like the Google Trends search-based index as a leading indicator for private
consumption (Choi and Varian, 2009; Ginsberg et al, 2009; Suhoy, 2009; and Vosen and
Schmidt, 2011). Swallow and Labbe (2013) show that the Google trends search results
provide the useful information about sales of automobiles in an emerging market. They show
that the models incorporating the Google Trends Automotive Index outperform benchmark
specification both in-sample and out-of-sample nowcasts. Further, Vosen and Schmidt (2011)
compared Google Trends search-based index to survey-based index like Index of Consumer
Sentiment from Michigan survey and Consumer Confidence Index from the Conference
Board and reported that all the Google Trends indicator outperforms the survey-based
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indicators in terms of forecast performance.
In this research, we explore the utility of ICS and the GTI in the Food and Beverage
CPI forecast models. Moreover, we compare the forecast performance of our models utilizing
ICS and GTI data and compare it with the forecasts released by the USDA Economic
Research Service. We find that not only are ICS and GTI significant determinants of future
food price changes, but that models incorporating these measures outperform USDA
forecasts.
Data
Food-Related Consumer Price Index
U.S. Bureau of Labor Statistics (BLS) reports the Consumer Price Index (CPI) as an
economic indicator, a deflator of other economic series, or a means of adjusting dollar values.
The CPI represents the average change in prices paid by urban consumers for a market basket
of goods and services over time. Urban consumers are divided into two groups in measuring
CPI process: all urban consumers, and urban wage earners and clerical workers. The first
group covers 87 percent of the total U.S. population such as the professionals, the self-
employed, the poor, and the unemployed. Since the subjects of this group is residents of
metropolitan area, the Consumer Price Index for all urban consumers (CPI-U) does not reflect
the spending pattern of people who live in rural nonmetropolitan areas like farm family. The
Consumer Price Index for urban wage earners and clerical workers (CPI-W) is index based
on the second group. To be considered as the second group, more than one-half of the
household’s income must come from clerical or wage occupations, and at least one of the
household’s earners must be employed for at least 37 weeks of the last 12 months. As a subset
of the first group, it covers around 32 percent of U.S population.
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The market basket of goods and services reflected in the CPI can be separated to
eight categories: Food and Beverages, Housing, Apparel, Transportation, Medical Care,
Recreation, Education and Communication, and Other goods and Services. In 2011-2012, the
relative importance of the Food and Beverage component in CPI-U was 14.9 out of 100.
This research investigates the movement of Food and Beverages CPI-U with reference base,
1982-84=100. We also investigate whether total CPI across eight categories is an exogenous
predictor of Food and Beverage CPI.
Search-based Index (Google Trends Index)
Google Trends provides a measure of the popularity of terms that google users have searched
over time. The index of Google Trends represents how many searches have been conducted
for a particular term, relative to the total number of searches done on Google over time.
Specifically,
(1) 𝐺𝑜𝑜𝑔𝑙𝑒 𝑡𝑟𝑒𝑛𝑑𝑠 𝐴𝑡 =𝑆𝐴𝑡
𝑀𝑎𝑥(𝑆𝐴1,𝑆𝐴2,…,𝑆𝐴𝑡)× 100
where 𝐺𝑜𝑜𝑔𝑙𝑒 𝑡𝑟𝑒𝑛𝑑𝑠 𝐴𝑡 is a percentage of certain term entered at t-th period, 𝑆𝐴𝑡 is the
absolute search numbers of term A at t-th period, and 𝑀𝑎𝑥(𝑆𝐴1, 𝑆𝐴2, … , 𝑆𝐴𝑡) is the highest
values among 𝑆𝐴𝑡. 𝐺𝑜𝑜𝑔𝑙𝑒 𝑡𝑟𝑒𝑛𝑑𝑠 𝐴𝑡 is presented on a scale from 0 to 100. In this study,
we creating an index based on the search term “food prices”. Google Trends Index is
available from January 2004 and the highest point of our data is May 2008.
Consumer Sentiment
There are several survey-based indices such as the Livingston survey and the survey of
professional forecasters (SPF). These indices are conducted twice a year, in June and
December, and the middle of every quarter, respectively. Also, both ask economists from
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industry, government, and academia. Unlike Livingston and SPF, the Index of Consumer
Sentiment from Michigan is measured monthly and participants are households. As such,
the ICS is likely a more appropriate index to apply consumer’s expectation and sentiment to
forecast food-related CPI.
The University of Michigan has reported monthly ICS data since 1978, and reference base
is March 1997. The ICS is derived from the following five questions
𝑄1. Personal Finance Current: We are interested in how people are getting along financially
these days. Would you say that you (and your family living there) are better off or worse off
financially than you were a year ago?
𝑄2. Personal Finance Expected: Now looking ahead- do you think that a year from now you
(and your family living there) will be better off financially, or worse off, or just about the
same as now?
𝑄3. Business Condition 12 Month: Now turning to business conditions in the country as a
whole- do you think that during the next twelve months we will have good times financially,
or bad times, or what?
𝑄4. Business Condition 5 years: Looking ahead, which would you say is more likely- that in
the country as a whole we will have continuous good times during the next five years or so,
or that we will have periods of widespread unemployment or depression, or what?
𝑄5. Buying Conditions: About the big things people buy for their homes-such as furniture, a
refrigerator, stove, television, and things like that. Generally speaking, do you think now is a
good or bad time for people to buy major household items?
Based on above questions, the ICS is calculated
(2) ICS =∑ 𝑄𝑖
5𝑖=1
6.7558+ 2.0
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where 𝑄𝑖 is the i-th index question, 6.7558 is total score based on1996, and 2.0 is a constant
to correct for sample design changes from the 1950s.
Methods
ARIMAX model
While the pure autoregressive integrated moving average model (ARIMA) is composed of
lagged dependent variables and errors, an autoregressive integrated moving average model
with exogenous variables model (ARIMAX) includes the dependent variable, lagged
dependent variable, and the other variables in the equation to explain the external effect on
the dependent variables.ARIMAX model assumes that the future value of a variable is a
linear functions of past observations and independent variables. The general ARIMAX (p,d,q)
process has the form
(3) ∆𝑦𝑡 = 𝜃0 + ∑ ∅𝑖∆𝑦𝑡−𝑖𝑝𝑖=1 + 𝜀𝑡 − ∑ 𝜃𝑘𝜀𝑡−𝑘
𝑞𝑘=1 + ∑ 𝜋𝑗∆𝑥𝑗𝑡−1
𝑠𝑗=1
where ∆𝑦𝑡 is the differenced time series values at time t, ∆𝑦𝑡−𝑖 denotes the differenced
previous values at time t-i, 𝜀𝑡 is random error which follows white noise process, ∆𝑥𝑗𝑡 is
the jth independent variable at time t-1, p is the number of auto-regression terms, q is the
number of moving- average terms, and 𝑠 is the number of exogenous variables.
In this research, CPI of all items (AllCPI), the Google Trend Index (GTI), and the
Index of Consumer Sentiment (ICS) are considered as the exogenous variables for the model.
Thus, the first specifications of the ARIMAX (p,d,q) model is
(4) ∆𝐹𝐶𝑃𝐼𝑡 = 𝜃0 + ∑ ∅𝑖𝑝𝑖=1 ∆𝐹𝐶𝑃𝐼𝑡−𝑖 + 𝜃1∆𝐴𝑙𝑙𝐶𝑃𝐼𝑡−1 + 𝜃2∆𝐺𝑇𝐼𝑡−1 + 𝜃3∆𝐼𝐶𝑆𝑡−1 +
+𝜀𝑡 − ∑ 𝜌𝑗𝜀𝑡−𝑗𝑞𝑘=1
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where ∆𝐹𝐶𝑃𝐼𝑡 is the first differenced Food and Beverage category’s Consumer Price Index,
∆𝐹𝐶𝑃𝐼𝑡−𝑖 is the first differenced ith lags of ∆𝐹𝐶𝑃𝐼𝑡, ∆𝐴𝑙𝑙𝐶𝑃𝐼𝑡−1 is the first differenced
Consumer Price Index about all items at time t-1, ∆𝐺𝑇𝐼𝑡−1 is the first differenced Google
Trends Index about “Food Prices” at time t-1, ∆𝐼𝐶𝑆𝑡−1 is the first differenced Index of
Consumer Sentiment at time t-1, and 𝜀𝑡 is the stochastic error term which is independently
and identically distributed with a mean of zero and constant variance of 𝜎2.
VAR and VARX Models
A vector autoregression (VAR) model is a multivariate extension of simple autoregressive
model. Sims (1980) proposed models where all variables are jointly endogenous and
symmetric. The main goal of VAR model is to determine the interrelationship among
variables. Thus, Sims (1980) and Sims, Stock, and Watson (1990) suggest the variables in
level are more approporiate than those of differencing, even if the variables are not stationary
over time. Of course, the VAR in first differences is possible if the variables are I(1) and not
cointegrated. However, if the variables are cointegrated, only the VAR in levels is
appropriate. The VAR model in standard form is followed;
(5) 𝑥𝑡 = 𝐴0 + ∑ 𝐴𝑖𝑥𝑡−𝑖𝑖 + 𝑒𝑡
where 𝑥𝑡 is a (n×1) vector containing each of the n variables included in the VAR, 𝐴0 is a
(n×1) vector of intercept terms, 𝐴𝑖 is (n×n) matrices of coefficients, and 𝑒𝑡 is a (n×1)
vector of error term.
Now consider a VAR model for this research:
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(6) [
𝑙𝑛𝐹𝐶𝑃𝐼𝑡𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡𝑙𝑛𝐺𝑇𝐼𝑡𝑙𝑛𝐼𝐶𝑆𝑡
] = [
𝛼1
𝛼2𝛼3
𝛼4
] +
[ 𝛼11
1 𝛼121 𝛼13
1 𝛼141
𝛼211 𝛼22
1 𝛼231 𝛼24
1
𝛼311
𝛼411
𝛼321
𝛼421
𝛼331
𝛼431
𝛼341
𝛼441 ]
[
𝑙𝑛𝐹𝐶𝑃𝐼𝑡−1
𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡−1
𝑙𝑛𝐺𝑇𝐼𝑡−1
𝑙𝑛𝐼𝐶𝑆𝑡−1
] + ⋯+
[ 𝛼11
𝑝 𝛼12𝑝 𝛼13
𝑝 𝛼14𝑝
𝛼21𝑝 𝛼22
𝑝 𝛼23𝑝 𝛼24
𝑝
𝛼31𝑝
𝛼41𝑝
𝛼32𝑝
𝛼42𝑝
𝛼33𝑝
𝛼43𝑝
𝛼34𝑝
𝛼44𝑝
]
[ 𝑙𝑛𝐹𝐶𝑃𝐼𝑡−𝑝
𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡−𝑝
𝑙𝑛𝐺𝑇𝐼𝑡−𝑝
𝑙𝑛𝐼𝐶𝑆𝑡−𝑝 ]
+ [
𝜀𝐹𝐶𝑃𝐼𝑡
𝜀𝐴𝑙𝑙𝐶𝑃𝐼𝑡𝜀𝐺𝑇𝐼𝑡
𝜀𝐼𝐶𝑆𝑡
]
where 𝑙𝑛𝐹𝐶𝑃𝐼𝑡, 𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡, 𝑙𝑛𝐺𝑇𝐼𝑡, and 𝑙𝑛𝐼𝐶𝑆𝑡 are stationary, 𝑎𝑖𝑗𝑘 are the autoregressive
coefficients i = 1,2,3,4, j=1,2,3,4 and k=1,2,…p, and 𝜀𝐹𝐶𝑃𝐼𝑡, 𝜀𝐴𝑙𝑙𝐶𝑃𝐼𝑡, 𝜀𝐺𝑇𝐼𝑡, and 𝜀𝐼𝐶𝑆𝑡 are
white-noise disturbance with the standard deviations of 𝜎𝐹𝐶𝑃𝐼, 𝜎𝐴𝑙𝑙𝐶𝑃𝐼, 𝜎𝐺𝑇𝐼, and 𝜎𝐼𝐶𝑆,
respectively.
To determine exogenous variables in vector autoregressive model with exogenous
variable (VAR-X), weak exogeneity test and Granger-causality test are conducted. The
standard VAR-X model is followed;
(7)𝑥𝑡 = 𝐴0 + ∑𝐴𝑖𝑥𝑡−𝑖
𝑝
𝑖=1
+ ∑𝐵𝑖𝑦𝑡−𝑖
𝑞
𝑖=1
+ 𝑒𝑡
where 𝑦𝑡 is a (n×1) vector of exogenous variables, 𝐵𝑖 is (n×n) matrices of coefficients, and
𝑒𝑡 is a vector of error term.
VECM and VECMX models
A vector error-correction (VECM) model indicates how short-term dynamics of variables in
the system are influenced by the discrepancies from the long-run equilibrium. In the equation,
the variables respond to previous period’s deviation from long-run equilibrium, the lagged
variables in change, and stochastic shocks. Since the left hand side of the equation is I(0), the
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right hand side should be I(0). That is, the linear combination of variables must be stationary.
The generalized n-variables VECM model is followed
(8) ∆𝑥𝑡 = 𝜋0 + 𝜋𝑥𝑡−1 + ∑ 𝜋𝑖∆𝑥𝑡−𝑖𝑖 + 𝑒𝑡
where 𝜋0 is a (n×1) vector of intercept terms with elements 𝜋𝑖0, 𝜋𝑖 is a (n×n) coefficient
matrices with elements 𝜋𝑗𝑘(𝑖), 𝜋 is a matrix with elements 𝜋𝑗𝑘 such that one or more of
the 𝜋𝑗𝑘 ≠ 0, and 𝑒𝑡 is a (n×1) vector with elements 𝑒𝑖𝑡.
As specified, the VECM model form for this research is followed
(9) [
∆𝑙𝑛𝐹𝐶𝑃𝐼𝑡∆𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡∆𝑙𝑛𝐺𝑇𝐼𝑡∆𝑙𝑛𝐼𝐶𝑆𝑡
] = [
𝛼1
𝛼2𝛼3
𝛼4
] + [
𝛾11 𝛾12 𝛾13 𝛾14
𝛾21 𝛾22 𝛾23 𝛾24
𝛾31
𝛾41
𝛾32
𝛾42
𝛾33
𝛾43
𝛾34
𝛾44
] [
𝑙𝑛𝐹𝐶𝑃𝐼𝑡−1
𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡−1
𝑙𝑛𝐺𝑇𝐼𝑡−1
𝑙𝑛𝐼𝐶𝑆𝑡−1
] +
[ 𝛼11
1 𝛼121 𝛼13
1 𝛼141
𝛼211 𝛼22
1 𝛼231 𝛼24
1
𝛼311
𝛼411
𝛼321
𝛼421
𝛼331
𝛼431
𝛼341
𝛼441 ]
[
∆𝑙𝑛𝐹𝐶𝑃𝐼𝑡−1
∆𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡−1
∆𝑙𝑛𝐺𝑇𝐼𝑡−1
∆𝑙𝑛𝐼𝐶𝑆𝑡−1
] + ⋯+
[ 𝛼11
𝑝−1 𝛼12𝑝−1
𝛼13𝑝−1 𝛼14
𝑝−1
𝛼21𝑝−1 𝛼22
𝑝−1𝛼23
𝑝−1 𝛼24𝑝−1
𝛼31𝑝−1
𝛼41𝑝−1
𝛼32𝑝−1
𝛼42𝑝−1
𝛼33𝑝−1
𝛼43𝑝−1
𝛼34𝑝−1
𝛼44𝑝−1
]
[ ∆𝑙𝑛𝐹𝐶𝑃𝐼𝑡−𝑝−1
∆𝑙𝑛𝐴𝑙𝑙𝐶𝑃𝐼𝑡−𝑝−1
∆𝑙𝑛𝐺𝑇𝐼𝑡−𝑝−1
∆𝑙𝑛𝐼𝐶𝑆𝑡−𝑝−1 ]
+ [
𝜀𝐹𝐶𝑃𝐼𝑡
𝜀𝐴𝑙𝑙𝐶𝑃𝐼𝑡𝜀𝐺𝑇𝐼𝑡
𝜀𝐼𝐶𝑆𝑡
]
The VECM model could be expressed with a multivariate VAR model in first
differences augmented by the error correction term when 𝛾𝑖𝑗 = 0. Therefore, at least of 𝛾𝑖𝑗
should not be zero. Like the VAR-X model, the exogenous variables for vector error
correction model with exogenous variable (VECM-X) are determined by weak-exogenous
test and Granger-causality test. The generalized form of VECM-X model is
(10) ∆𝑥𝑡 = 𝜋0 + 𝜋𝑥𝑡−1 + ∑ 𝜋𝑖∆𝑥𝑡−𝑖𝑝−1𝑖=1 ∑ ∅𝑖𝑦𝑡−𝑖
𝑞𝑖=1 + 𝑒𝑡
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where 𝑦𝑡 is a (n×1) vector of exogenous variables, ∅𝑖 is a (n×n) coefficient matrices with
elements ∅𝑗𝑘(𝑖), and 𝑒𝑡 is a (n×1) vector with elements 𝑒𝑖𝑡.
Unit Root Tests
The unit root tests such as Augmented Dickey-Fuller(ADF) test, Phillips-Perron(PP) test,
Dickey-Fuller generalized least squares(DF-GLS) test, KPSS test, Park’s J test and Park’s G
test could be applied to identify the stationary of the data. In this research, the ADF test is
conducted.
The three version of ADF test is
(10) 𝑦𝑡 = 𝜌𝑦𝑡−1 + 𝜀𝑡
(11) 𝑦𝑡 = 𝛼 + 𝜌𝑦𝑡−1 + 𝜀𝑡
(12) 𝑦𝑡 = 𝛼 + 𝛽𝑡 + 𝜌𝑦𝑡−1 + 𝜀𝑡
where 𝜀𝑡 follows I(0) with 0 mean. Equation (10) is the test for unit root without drift,
equation (11) is the test for unit root with drift and Equation (12) is the test for unit root with
drift and a deterministic trend. The null hypothesis of unit root is 𝐻0: 𝜌 = 0. Under this null
hypothesis, we compute 𝜏𝜇 and 𝜏𝑡, then modify them to get 𝑧𝜇 and 𝑧𝑡, of which
asymptotic distribution is the DF distribution for 𝜏𝜇 and 𝜏𝑡.
If the variable is stationary over time, autoregressive (AR), autoregressive moving
average (ARMA), and vector autoregression (VAR) model would be applied. As for the
nonstationary data, if cointegration exists, we could estimate vector error correction (VECM)
model. On the other hand, when there is no cointegration relationship among variables, both
VAR in level and VAR in difference could be applied.
Johansen’s Cointegration Rank Test
13
Engle and Granger (1987) introduced the concept of co-integration. The basic idea is
considering a set of multiple nonstationary time-series variables in the long-run equilibrium.
This long-run relationship between variables describes how variables adjust to deviation from
equilibrium. There are two conditions for cointegration. The component of vector 𝑦𝑡 =
(𝑦1𝑡, 𝑦2𝑡, … , 𝑦𝑛𝑡)′ are said to be cointegrated of order d,b, if first, all components of 𝑦𝑡 are
integrated of order d. Second, there exists a cointegrating vector β = (𝛽1, 𝛽2, … , 𝛽𝑛) such
that the linear combination β𝑦𝑡 = 𝛽1𝑦1𝑡 + 𝛽1𝑦1𝑡 + ⋯+ 𝛽1𝑦1𝑡 is integrated of order (d-b)
where b>0. Also, the number of cointegrating vectors is called the cointegrating rank of 𝑦𝑡. If
𝑦𝑡 has n components, n-1 linearly independent cointegrationng vectors could exist at most.
Thus, in this research, the maximum rank number of cointegration vectors is 3.
Engel and Granger (1987) method has several defects. First, it relies on two step
estimator. Thus, step 1 error is carried into step 2. Also, this method is not appropriate to
apply to three or more variables case. The estimation requires that one variable should be
placed on the left-hand side and others must be used as regressors. However, in the
multivariate case, any of the variables can be placed on the left hand side. Johansen (1988)
procedure circumvents several defects of Engel and Granger (1987) procedure. So, it could
avoid two-step estimation problems and be applied to estimation and testing for the multiple
co-integration vectors.
Johansen suggested two test statistics to test the null hypothesis that there are at most
r cointegration vectors.
𝐻0: 𝑟𝑎𝑛𝑘(𝜋) ≤ 𝑟 𝑜𝑟 𝜋 = 𝛼𝛽′
where α is the speed of adjustment coefficients and β is long-run parameter (p × r)
matrices, p is a sequence {𝜀𝑡} of i.i.d dimension, and r is rank. A large value of the speed of
adjustment coefficient implies that the variable is greatly responsive to last period’s
14
equilibrium error. Though two rank test shares the same null hypothesis, the alternative
hypothesis is different from each other. As for the trace test, the alternative hypothesis is
𝐻1: 𝑟𝑎𝑛𝑘(𝜋) > 𝑟
And the trace statistics is followed
(13) 𝜆𝑡𝑟𝑎𝑐𝑒 = −𝑇 ∑ log(1 − 𝜆𝑖)𝑝𝑖=𝑟+1
where 𝜆𝑖 are the p-r smallest squared canonical correlations.
In the case of the maximum eigenvalue test, the alternative hypothesis and statistic
are followed
𝐻1: 𝑟𝑎𝑛𝑘(𝜋) ≥ 𝑟 +1
(14) 𝜆𝑚𝑎𝑥 = −𝑇 log(1 − 𝜆𝑟+1)
These test results could conflict each other. As such the maximum eigenvalue test is
considered as having the sharper alternative hypothesis.
Granger-Causality Test and Weak Exogeneity Test
Granger-causality test refers the effects of past and a current value of one variable on the
current value of another variable. Suppose the 𝑥𝑡 vector in equation (5) is (𝑦𝑡 𝑧𝑡)′. If the
past and current value of {𝑦𝑡} could improve the forecasting performance of {𝑧𝑡}, then we
could say 𝑦𝑡 Granger cause 𝑧𝑡. Thus the null hypothesis of Granger-Causality test is
𝐻0: 𝑎21 = 0
where 𝑎21 is the autoregressive coefficient.
Since the variables are I(1), the chi-squared (Wald) test is more appropriate rather
than t-test or F-test. When the null hypothesis could be rejected, there exists a Granger-
causality relationship. As such, the Granger-causality test is different from an exogeneity test.
While the Granger-causality test is about the effects of past values of a certain variable on the
15
current values of the other, the exogeneity test deal with the effects of a contemporaneous
value of 𝑦𝑡 on 𝑧𝑡. However, in multivariate case, the Granger-causality restriction imply a
weak exogeneity form.
In a cointegrated process, the interpretation of Granger-causality test is different from
usual case. Again, let’s suppose the 𝑥𝑡 vector in equation (8) is (𝑦𝑡 𝑧𝑡)′. If lagged values
∆𝑦𝑡−𝑖 is not included in the ∆𝑧𝑡 equation and if 𝑧𝑡 does not respond to the discrepancy
from long-run equilibrium, then we could say that {𝑦𝑡} does not Granger cause {𝑧𝑡}. As for
the weak exogeneity test, if the speed of adjustment parameter of 𝑧𝑡 is zero, we could
conclude that 𝑧𝑡 is weakly exogenous.
Forecast encompassing test
There are two well-known forecast encompassing tests. Fair and Shiller(1989) considered the
ability of forecasts to explain the original value. On the other hand, Chong and Hendry
(1986) tested one forecast to describe the error term of the other forecast. In this research, the
Fair and Shiller test employed. The equation is blow:
(15) 𝐹𝐶𝑃𝐼𝑡 = 𝛼 + 𝜆1𝑓1𝑡 + 𝜆2𝑓2𝑡 + 𝑣𝑡
where 𝐹𝐶𝑃𝐼𝑡 is the real value of Food and Beverage CPI, 𝑓𝑖𝑡 is the ith forecast value where
i=1,2, 𝜆𝑖 is the coefficients of ith forecast, and 𝑣𝑡 is the error term.The null hypothesis is
𝐻0: 𝜆1 = 0, then the equation is
(16) 𝐹𝐶𝑃𝐼𝑡 = 𝛼 + 𝜆2𝑓2𝑡 + 𝑣𝑡
Also, in the case of the alternative hypothesis which is 𝐻1: 𝜆2 = 0, the equation is
(17) 𝐹𝐶𝑃𝐼𝑡 = 𝛼 + 𝜆1𝑓1𝑡 + 𝑣𝑡
Based on the significance of 𝜆𝑖, if we could reject 𝐻0: 𝜆1 = 0 and fail to reject
𝐻1: 𝜆2 = 0, then it indicates that redundancy of 𝑓2𝑡. That is, the 𝑓1𝑡 forecast encompasses
16
the 𝑓2𝑡 forecast. In the same vein, for switched the null and alternative hypothesis, the
interpretation is in opposite direction. Also, in the case when both null and alternative
hypothesis are rejected at the same time, it indicates that the combined (weighted) forecast
with 𝑓1𝑡 and 𝑓2𝑡 provide the better forecast.
Results
Tests for Estimating Food and Beverage CPI forecast model
Unit root test
To find the Food and Beverage CPI forecast model, the Augmented Dickey-Fuller unit root
test, Johansen’s cointegration test, weak exogeneity test and Granger-causality test are
conducted in this research. First of all, the Augmented Dickey-Fuller unit root test identify
whether the variables are stationary over time. The general to specific methodology (t-test)
and measurement of model selection which is Akaike Information Criteria (AIC) and
Schwarz Bayesian Criterion (SBC) are considered as the criteria to select the optimal lag for
the unit root test. In the case where the results are different from each other, we choose the
lag which is proved at least two criteria. As for FCPI in level, FCPI in difference, AllCPI in
difference and ICS in difference, the result of general to specific test are consistent with that
of AIC and SBC. On the other hand, AllCPI in level, GTI in level, ICS in level, and GTI in
difference do not have same results between criteria. For the AllCPI in level, second lag is
selected as the optimal lag by t-test and SBC. And fifth, third, and sixth lag are chosen by t-
test and AIC for GTI in level, ICS in level, and GTI in difference, respectively.
Table 2 denotes that the result of the Augmented Dickey-Fuller unit root test. We fail
to reject the null hypothesis of unit root for the variables in level at 1% significance level and
the null hypothesis of unit root for the first differenced variables are rejected at 5% level,
17
which means that the variables taking the first difference do not have the unit roots. Thus, we
could obtain the stationary variables using the first integration.
Johansen’s cointegration test
Because the variables are non-stationary over time and share the first order of integration,
Johansen’s cointegration rank test is conducted to determine whether the long-run
equilibrium relationship exists between variables. Based on both trace and maximum
eigenvalue tests, we fail to reject the null hypothesis of two cointegration at 5% level. Table 4
indicates that the long run parameter β and the adjustment coefficient α with ln 𝐹𝐶𝑃𝐼 as
normalized in the case where four variables are cointegrated order 2.
Based on the variables’ nonstationarity and cointegration property, the VAR model in
level and the VECM model are proposed to forecast Food and Beverage CPI. The long-run
equilibrium relationship in VECM model is determined by cointegration vector (π = α𝛽′).
Since, other terms except for the error correction term are stationary, to make left-hand side
stationary, the error correction term should be stationary, which is correspond with the
definition of cointegration.
Granger-Causality test and Weak exogeneity test
For the VAR model, the results of Granger-causality test directly indicates that of weak
exogeneity test. Table 5 shows that test 1 and test 3 reject the null hypothesis at 1%
significance level, which means that group 1 variables (ln 𝐹𝐶𝑃𝐼 and ln 𝐺𝑇𝐼) are influenced
by group 2 variables (Other variables except for ln 𝐹𝐶𝑃𝐼 and ln 𝐺𝑇𝐼, respectively). On the
other side, ln 𝐴𝐶𝑃𝐼 and ln 𝐼𝐶𝑆 does not Granger cause by other variables. Based on this
results, ln 𝐴𝐶𝑃𝐼 and ln 𝐼𝐶𝑆 are chosen as the exogenous variables in the VAR-X model.
18
Table 6 shows the results of weak exogeneity test for the VECM model. ln 𝐹𝐶𝑃𝐼
and ln 𝐺𝑇𝐼 reject null hypothesis of a weak exogenous variable at 1% level, and ln 𝐴𝐶𝑃𝐼
reject at 5% level. ln 𝐼𝐶𝑆 fails to reject the null hypothesis, which means we could consider
ln 𝐼𝐶𝑆 as the exogenous variable in the VECM-X model. To define the exogenous variable in
the VECM-X model, the result of Granger-causality test also should be considered. Since test
1 and test 3 reject the null hypothesis at 1% level, we could define ln 𝐹𝐶𝑃𝐼 and ln 𝐺𝑇𝐼 are
as the endogenous variables and ln 𝐴𝐶𝑃𝐼 and ln 𝐼𝐶𝑆 are as the exogenous variable.
Considering both results, ln 𝐴𝐶𝑃𝐼 and ln 𝐼𝐶𝑆 that fail to reject at 1% level in both tests are
designated as the exogenous variable in the VECM-X model.
Comparing to forecasting performance
The forecast models are estimated using in sample data from January 2004 to December 2009
which is six years. Table 7 denotes the results of assessing the predictive performance of the
forecast models; the root mean square error (RMSE) and mean absolute percentage error
(MAPE) of each forecasting models about the period from January 2010 to July 2015. We
conclude that the model which has the smallest RMSE and MAPE provides the best forecast
values compared to other models. According to Table 7, VAR model outperforms ARIMA-
X,VAR-X, VECM and VECM-X models.
United States Department of Agriculture Economic Research Service (USDA ERS)
has reported Food CPI forecasts (Kuhns et al., 2015). For the forecast of Food CPI’s
subcategories, the vertical price transmission ECM approach and the autoregressive moving-
average approach are used. The selection of the methodology depends on data availability. If
we could get sub-category’s information of multiple stages involved in the U.S. food supply
system and the food categories’ data are cointegrated order r, then the vertical price
19
transmission ECM methodology was applied. On the other hand, if there is such data
limitation about sub-categories, the traditional forecast model, the autoregressive moving-
average approach are used. To get the forecasts for larger categories, USDA conducted the
weighted average of the forecasted subcategories’ CPI.
USDA have updated Food CPI data as much as they could apply to estimate forecast
of CPI. This method seems similar to rolling window method. The difference between two
methods is the length of window. The classic rolling window method has constant the length
of window over time. However, the window of USDA method is increased over time. To
compare our suggested VAR model with USDA model, we measured the first forecast values
using five years monthly data (total 60 data) and for the next forecast values five years
monthly data plus the next month data (total 61 data) were used. Table 8 indicates that the
suggested model has the smaller value of RMSE and MAPE than those of USDA.
Forecast Encompassing Test
The suggested VAR forecast value is employed as 𝑓1𝑡 in equation (15) and as a benchmark
model(𝑓2𝑡), the forecast of USDA ERS is considered. Table 9 indicates that we reject the null
hypothesis of 𝐻0: 𝜆1 = 0 and fail to reject the alternative hypothesis of 𝐻1: 𝜆2 = 0, which
means that the VAR forecast information encompass the USDA forecast information. This
result has same implication with the results of rolling window method.
Conclusions
We examined whether unconventional consumer-oriented measures might be useful in the
predicting Food and Beverages Consumer Price Index (CPI). According to out of sample
forecasts, we found that our best fitting model was a VAR, which includes measures related
20
to the Index of Consumer Sentiment (ICS) from Michigan survey and a Google Trends Index
based on food prices. Moreover, we found that our model with the consumer-oriented
measures outperforms the USDA model in predicting Food and Beverage CPI.
21
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Table 1. Information criterial for selection of optimal lag for unit root test
Variables lag AIC SBC Variables lag AIC SBC
log(𝐹𝑜𝑜𝑑𝐶𝑃𝐼) 6 -1273.07 -1252.53 ∆ log(𝐹𝑜𝑜𝑑𝐶𝑃𝐼) 6 -1279.51 -1259.02
5 -1264.18 -1246.57 5 -1281.51 -1263.95
4 -1263.91 -1249.24 4 -1282 -1267.37
3 -1233.99 -1222.25 3 -1282.92 -1271.21
2 -1222.62 -1213.82 2 -1274.7 -1265.92
1 -1189.5 -1183.63 1 -1276.1 -1270.25
log(𝐴𝑙𝑙𝐶𝑃𝐼) 6 -1137.55 -1117 ∆ log(𝐴𝑙𝑙𝐶𝑃𝐼) 6 -1155.08 -1134.59
5 -1140.14 -1122.54 5 -1156.75 -1139.18
4 -1143.14 -1128.46 4 -1155.28 -1140.64
3 -1145.49 -1133.75 3 -1157.09 -1145.38
2 -1145.1 -1136.3 2 -1158.38 -1149.6
1 -1089.2 -1083.33 1 -1151.14 -1145.28
log(𝐺𝑇𝐼) 6 -90.4766 -69.9353 ∆ log(𝐺𝑇𝐼) 6 -90.1983 -69.7075
5 -92.1188 -74.5119 5 -84.8367 -67.2732
4 -80.9795 -66.3071 4 -86.8324 -72.1961
3 -80.5345 -68.7966 3 -68.997 -57.288
2 -82.5342 -73.7308 2 -61.6852 -52.9035
1 -80.6723 -74.8033 1 -60.1338 -54.2793
log(𝐼𝐶𝑆) 6 -394.029 -373.488 ∆ log(𝐼𝐶𝑆) 6 -390.08 -369.59
5 -394.754 -377.147 5 -391.693 -374.129
4 -396.739 -382.067 4 -392.3 -377.664
3 -396.964 -385.226 3 -394.273 -382.564
2 -391.767 -382.964 2 -394.371 -385.589
1 -393.658 -387.789 1 -388.71 -382.855
25
Table 2. Augmented Dickey-Fuller Test Unit Root Test
Variables Optima
l lags
Zero mean Single mean Trend
𝜏𝜇 𝑃𝑟 < 𝜏𝜇 𝜏𝜇 𝑃𝑟 < 𝜏𝜇 𝜏𝜇 𝑃𝑟< 𝜏𝜇
log(𝐹𝑜𝑜𝑑𝐶𝑃𝐼) 6 2.2137 0.9936 -1.0114 0.7474 -2.2077 0.4811
log(𝐴𝑙𝑙𝐶𝑃𝐼) 2 3.0079 0.9993 -1.5491 0.5061 -2.7115 0.2337
log(𝐺𝑇𝐼) 5 0.0617 0.7013 -2.4393 0.1330 -3.2857 0.0732
log(𝐼𝐶𝑆) 3 -0.1025 0.6472 -1.8069 0.3758 -1.5175 0.8188
∆ log(𝐹𝑜𝑜𝑑𝐶𝑃𝐼) 3 -2.3476 0.0188 -3.6444 0.0061 -3.6963 0.0259
∆ log(𝐴𝑙𝑙𝐶𝑃𝐼) 2 -5.6036 <.0001 -6.5847 <.0001 -6.7068 <.0001
∆ log(𝐺𝑇𝐼) 6 -4.9666 <.0001 -4.9509 0.0001 -4.9405 0.0005
∆ log(𝐼𝐶𝑆) 2 -8.6271 <.0001 -8.5945 <.0001 -8.7171 <.0001
26
Table 3. Johansen’s cointegration rank tests
Trace Test
𝐻0: 𝑅𝑎𝑛𝑘 = 𝑟 𝐻0: 𝑅𝑎𝑛𝑘 > 𝑟 Trace Statistics 5% Critical Value
0 0 78.6566 47.21
1 1 31.7741 29.38
2 2 10.5880 15.34
3 3 1.3071 3.84
Maximum Eigenvalue Test
𝐻0: 𝑅𝑎𝑛𝑘 = 𝑟 𝐻0: 𝑅𝑎𝑛𝑘 = 𝑟 + 1 Max Statistics 5% Critical Value
0 1 46.8825 27.07
1 2 21.1862 20.97
2 3 9.2809 14.07
3 4 1.3071 3.76
Table 4. Long-run parameter 𝛃 estimates and adjustment coefficient 𝛂 estimates
(rank=2)
Long-run β Adjustment coefficient α Variable 1 2 1 2
ln 𝐹𝐶𝑃𝐼 1.00000 1.00000 -0.05083 -0.024
ln 𝐴𝐶𝑃𝐼 -0.06888 0.05931 6.04234 -2.41840
ln 𝐺𝑇𝐼 -0.03810 0.25926 -0.55104 -0.63909
ln 𝐼𝐶𝑆 -1.19815 -0.54352 0.01165 0.05409
27
Table 5. The results of Granger-causality Test
Tests VAR VECM
Optimal
Lag 𝜒2 Pr > 𝜒2 Optimal
Lag 𝜒2 Pr > 𝜒2
1 2 22.76 0.0009** 2 24.36 0.0004**
2 2 10.96 0.0897 2 11.73 0.0683
3 2 28.92 <.0001** 2 30.95 <.0001**
4 2 6.19 0.4025 2 6.62 0.3571 **Significant at the 0.01 level and *significant at the 0.05 level.
Test 1: Group 1 is ln 𝐹𝐶𝑃𝐼 and Group 2 is ln 𝐴𝐶𝑃𝐼, ln 𝐺𝑇𝐼, ln 𝐼𝐶𝑆.
Test 2: Group 1 is ln 𝐴𝐶𝑃𝐼 and Group 2 is ln 𝐹𝐶𝑃𝐼, ln 𝐺𝑇𝐼, ln 𝐼𝐶𝑆.
Test 3: Group 1 is ln 𝐺𝑇𝐼 and Group 2 is ln 𝐹𝐶𝑃𝐼, ln 𝐴𝐶𝑃𝐼, ln 𝐼𝐶𝑆.
Test 4: Group 1 is ln 𝐼𝐶𝑆 and Group 2 is ln 𝐹𝐶𝑃𝐼, ln 𝐺𝑇𝐼, ln 𝐴𝐶𝑃𝐼.
Table 6. The results of weak exogeneity test for the VECM model
Variable 𝜒2 Pr > 𝜒2
ln 𝐹𝐶𝑃𝐼 18.49 <.0001**
ln 𝐴𝐶𝑃𝐼 6.46 0.0395*
ln 𝐺𝑇𝐼 43.61 <.0001**
ln 𝐼𝐶𝑆 0.43 0.8046
**Significant at the 0.01 level and *significant at the 0.05 level.
28
Table 7. Forecasting Comparison using RMSE and MAPE (Out of sample)
Models RMSE MAPE
ARIMA-X(3,1,0) 0.01365 0.20882445
VAR(2) 0.00510 0.0753417
VAR-X(1,1) 0.00734 0.11230649
VECM(2) 0.00648 0.09925073
VECM-X(1,2) 0.00553 0.0854346
Table 8. Forecasting Comparison with USDA
Models RMSE MAPE
USDA model 0.02381 0.29728432
VAR(2) 0.0006719 0.00882988
Table 9. Encompassing test
Models t-value Pr >t
USDA model -1.52 0.2040
VAR(2) 88.91 <.0001 **Significant at the 0.01 level and *significant at the 0.05 level.