Relative prices of food and return volatility of
agricultural commodities: Evidence from some
latin american economies and India
Carlos Martins-Filho1 and Maximo Torero2
1
University of Colorado - Boulder and IFPRI
2
IFPRI.
July 7, 2014
The Initial Question
I
Is there empirical evidence of the existence of a correlation between
price volatility of major agricultural commodities and consumer
welfare?
Problems:
I
Changes in consumer welfare due to variations in (own) prices levels
are notoriously difficult to measure due to income effects associated
with price changes.
I
Prices of other goods in a consumption set are not fixed.
I
It is not uncommon in developing countries for consumers to be
producers of some of the food items in their consumption set.
I
Models for the dynamic evolution of (conditional) volatility of
agricultural commodities are often based on restrictive stochastic
models.
A Related Question
I
Is there empirical evidence of the existence of a correlation between
price volatility of major agricultural commodities and the change in
relative prices of certain defined food groups?
A clarification:
I There is empirical evidence of a positive correlation between changes
in prices levels of agricultural commodities and changes in relative
prices of certain food groups.
I The question being asked here is whether or not the volatility of
price changes has an impact in relative prices of certain food groups.
How to measure relative prices of food groups?
I
Let there be N goods and services. A consumption basket in time
period t = 0, 1, · · · , T and its price are denoted by
qt0 = qt1 · · · qtN , pt0 = pt1 · · · ptN .
The share of expenditures on element F of the basket in time period
t is
ptF qtF
stF = 0
pt qt
The Laspeyres index
I
The Laspeyres index from time period t − 1 to time period t is
L(pt , pt−1 , qt−1 ) =
I
The relative share of the price index change associated with element
F of the basket is
YtF =
I
N
X
ptn
st−1,n for t = 1, · · · , T
p
t−1,n
n=1
ptF
pt−1,F st−1,F
L(pt , pt−1 , qt−1 )
∈ (0, 1) for t = 1, · · · , T .
If YtF is close to 1 at time t, the element F in the consumption
basket accounts for a large share of price index variability.
A Model of Volatility for Agricultural Commodities
Let Pt be the price of an agricultural commodity at time t and
rt = log
Pt
Pt−1
be net returns. We assume the following conditional location-scale model
rt = m0 +
L
X
j=1

mj (rt−j ) + h0 +
L
X
1/2
hj (rt−j )
εt
(1)
j=1
where
I
L ∈ N, εt ∼ IID(0, 1) (in this paper L = 2)
I
E (mj (rt−j )) = E (hj (rt−j )) = 0 for all j, h0 > 0
Estimation of mj and hj for j = 0, 1, · · · , L is conducted as proposed in
Martins-Filho et al. (2013, 2014) using daily returns.
Volatility Estimation
I
We first estimate m0 , m1 and m2 . The estimation has two steps:
1. Pilot estimators for m0 , m1 and m2 are obtained using B-splines.
2. A one step back-fitting procedure based on a local linear estimation
is used to obtain final estimators m̂0 , m̂1 and m̂2 .
I
Next we define residuals ût = rt − m̂0 +
t = 3, · · · , T and estimate
ût2 = h0 +
2
X
P2
j=1
m̂j (rt−j ) for
hj (rt−j ) + νt
j=1
I
using the same two step procedure used to estimate the location.
√
The resulting ĥ0 , ĥ1 , ĥ2 are ThT asymptotically normal.
I
An estimated sequence of conditional volatilities is defined as

σ̂t = ĥ0 +
2
X
j=1
1/2
ĥj (rt−j )
for t = 3, · · · , T .
A General Stochastic Model for the Conditional
Expectation of YtF
In its most general form, we are interested in the estimation of
E (YtF |h1/2 (rt−1 , · · · , rt−L ), Wt ) = g −1 (m(h1/2 (rt−1 , · · · , rt−L ), Wt ))
for t = L + 1, · · · , T , where
I
Wt ∈ RK is a collection of suitably defined conditioning variables
I
g is a strictly monotonic link function g (x) : [0, 1] → R
I
m is a smooth function m(x) : RK +1 → R
The fact that YtF ∈ (0, 1) has important implications for stochastic
modeling.
Beta Regression
The Beta density is given by
π(y ; p, q) =
If µ =
p
p+q
Γ(p + q)
for p, q > 0, 0 < y < 1.
Γ(p)Γ(q)y p−1 (1 − y )q−1
and φ = p + q, then
E (Y ) = µ, V (Y ) =
µ (1 − µ)
.
1+φ
φ is a “precision” parameter. For fixed µ, a larger φ gives smaller
variance V (Y ).
Beta Regression
We consider a conditional Beta density where µ is such that
m(µt ) = α h1/2 (rt−1 , · · · , rt−L ) +
K
X
Wtj βj = Xt
j=1
and
g (µ) = log
α
β
= Xt θ
µ
, note that g −1 is the logistic function.
1−µ
This gives,
E (YtF |h1/2 (rt−1 , rt−2 ), Wt ) =
and
V (YtF |h1/2 (rt−1 , rt−2 ), Wt ) =
exp(Xt θ)
1 + exp(Xt θ)
E (YtF |·)(1 − E (YtF |·)
1+φ
Maximum Likelihood Estimation
The log-likelihood function based on a sample of size T is,
`(α, β, φ) =
T
X
`t (µt , φ)
t=1
with score vectors given by
`α,β (α, β, φ) = φX 0 D(Y ∗ − µ∗ )
`φ (α, β, φ) =
T
X
∗
(µt (YtF
− µ∗t ) + log (1 − YtF ) − ψ((1 − µt )φ)
t=1
+ψ(φ))
YtF
where Y ∗ has t th element Yt∗ = log 1−Y
, µ∗ has t th element
tF
µt
∗
µt = log 1−µt , ψ(·) is the digamma function,
0
D = diag {1/g 0 (µt )}T
t=1 , X =
X10
···
XT0
Maximum Likelihood Estimation
The Beta regression model satisfies standard regularity conditions for
asymptotic normality of ML estimators. We have,
√
φ
φ̂
d
−
→ N (0, K −1 )
T
θ
θ̂
where
K = −E
∂2
∂φ∂φ `(φ, θ)
∂2
∂θ∂φ `(φ, θ)
∂2
∂φ∂θ `(φ, θ)
∂2
∂θ∂θ `(φ, θ)
!
The impact of changes in covariate values
I
It is easy to show that
PK
W
β
exp
α
σ
+
tj
j
t
j=1
∂
E (YtF |σt , Wt ) = α
PK
∂σt
1 + exp α σt + j=1 Wtj βj
where σt = h1/2 (rt−1 , · · · , rt−L ).
I
The left-hand side should be interpreted as the impact that changes
in commodity volatility have on the share of aggregate price changes
associated with commodity basket item(s) F .
I
Such impact changes with t.
Data
I
Latin american countries: Costa Rica, El Salvador, Guatemala,
Honduras, Ecuador, Peru, Mexico, Nicaragua, Panama, Dominican
Republic
I
India
I
The length of the time series for each country is different
Food groups:
I
1.
2.
3.
4.
I
Bread and Cereals
Meat
Dairy products and eggs
Other food items
Covariates
1.
2.
3.
4.
Monthly index of economic activity: a Laspeyres Index
Imports
Oil prices
Measures of monthly volatility (average, median, interquartile range
of daily volatility)
Data
I
Monthly index of economic activity: This a Laspeyres index. It
measures the evolution of the economic activity, approximating the
aggregated value of the industries included in the calculation of the
GDP.
n
X
It =
Iit wi0
i=1
where:
I
I
I
I
It is the general index in period t
Ii t is the index of industry i (manufacturing, agricultural, etc) in
month t
wi0 is the weight that corresponds to industry i in the calculation of
GDP in the baseline period. n is the number of industries.
Monthly value of imports in millions of (constant) USD
Data
I
Volatility: We consider returns on future contract prices closest to
maturity for wheat (CBOT), wheat (KCBT), corn, soybeans and rice
from 01/28/1987 - 08/20/2013.
After obtaining the estimated daily volatilities, three indicators were
constructed to be used as control variables: i) Monthly means, ii)
Monthly medians, and iii) Montlhy inter-quantile ranges (0.25
percentile - 0.75 percentile).
I
Oil: Monthly oil prices were obtained from U.S. Energy Information
Administration
Regressand and Regressor of Interest
Regressand: Share of the change in the Laspeyres Index associated with
element F in a consumption basket.
Regressor: Volatility of various commodities.
Table : Economic Activity
Country
Ecuador 1
2
El Salvador 1
2
Guatemala 1
2
Honduras 1
2
Nicaragua 1
2
Panama 1
2
Peru 1
2
Breads
+
+
+
+
-
Meat
Dairy
Other Food
-
+
+
+
+
-
-
-
+
+
-
+
+
+
-
+
+
-
Table : Imports
Country
Ecuador 1
2
El Salvador 1
2
Guatemala 1
2
Honduras 1
2
Nicaragua 1
2
Panama 1
2
Peru 1
2
Breads
+
+
+
+
+
Meat
+
+
-
+
+
+
+
+
+
+
+
+
Dairy
+
+
+
+
+
+
Other Food
+
+
+
+
+
+
+
+
+
+
Table : Impact of Wheat Volatility on Breads and Cereals: * indicates
significant at the 0.95 level
Country
Ecuador 1
2
El Salvador 1
2
Guatemala 1
2
Honduras 1
2
Nicaragua 1
2
Panama 1
2
Peru 1
2
Coefficient sign
θ7 < 0, θ8∗ > 0
θ7 < 0, θ8∗ > 0
θ7 > 0, θ8∗ > 0
θ7∗ < 0, θ8∗ > 0
θ7 < 0, θ8 > 0
θ7∗ < 0, θ8∗ > 0
θ7∗ > 0, θ8 > 0
θ7∗ > 0, θ8∗ > 0
θ7 > 0, θ8∗ > 0
θ7 < 0, θ8 > 0
θ7 > 0, θ8 < 0
θ7∗ > 0, θ8 > 0
θ7 < 0, θ8∗ > 0
θ7 < 0, θ8∗ > 0
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : El Salvador
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
5417.5179
-2.1058
-0.0012
0.0001
0.026
0.4545
3.9607
-6.1587
3.6475
4.6638
t-Statistic
8.8879*
-43.6984*
-3.5698*
2.2854*
0.5287
0.3235
1.7053
-5.4004*
1.1918
1.7920
Pseudo-R 2
0.53
n=158
Marginal impact
-0.1966
-0.0001
0.0000
0.0025
0.0424
0.3697
-0.5749
0.3405
0.4354
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : El Salvador
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
5497.1816
-2.1150
-0.0010
0.0000
0.0297
1.1981
2.7575
-5.9155
-0.8470
8.9420
t-Statistic
8.8879*
-42.295*
-3.1781*
1.7162
0.5981
0.8516
1.2040
-5.2472*
-0.2366
3.4872*
Pseudo-R 2
0.84
n=158
Marginal impact
-0.1974
-0.0001
0.0000
0.0027
0.1118
0.2574
-0.5522
-0.0790
0.8348
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Guatemala
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
1716.9728
-3.1144
0.0036
0.0007
0.2001
-6.6993
10.8913
-11.7421
-7.2937
13.061
t-Statistic
6.5952*
-22.2008*
2.2634*
8.6971*
1.6149
-1.5239
2.4565*
-3.2729*
-0.85429
1.2745
Pseudo-R 2
0.94
n=87
Marginal impact
-0.3371
0.0003
0.0000
0.0216
-0.7253
1.1792
-1.2713
-0.7896
1.4141
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Guatemala
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
1934.4963
-3.1446
0.0030
0.0008
0.2192
-7.8565
10.6188
-4.5568
-30.2554
19.1239
t-Statistic
6.5952*
-22.9536*
1.9401*
9.8748*
1.8921
-1.8933
2.5663*
-1.2849
-3.7944*
1.9129*
Pseudo-R 2
0.95
n=87
Marginal impact
-0.3404
0.0003
0.0000
0.0237
-0.8506
1.1496
-0.4933
-3.2757
2.0705
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Honduras
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
16070.7240
-2.4801
-0.0045
0.0006
-0.0171
-1.3099
-4.079
2.6122
9.2651
0.4945
t-Statistic
6.9278*
-54.0230*
-8.2841*
7.1886*
-0.3559
-0.7635
-2.0809*
2.1661*
2.8726*
0.1127
Pseudo-R 2
0.78
n = 96
Marginal impact
-2.4801
-0.0002
0.0000
-0.0009
-0.0735
-0.2291
0.1467
0.5203
0.0277
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Honduras
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
14103.0940
-2.4861
-0.0046
0.0006
-0.0973
-0.75051
-2.5590
2.3909
9.4019
2.7155
t-Statistic
6.9278*
-47.0650*
-7.5630*
6.9350*
-0.7774
-0.4047
-1.2205
1.8151
2.1113*
0.5225
Pseudo-R 2
0.75
n = 96
Marginal impact
-0.1396
-0.0002
0.0000
-0.0022
-0.0421
-0.1437
0.1342
0.5280
0.1525
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Nicaragua
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
17216.9150
-2.8381
0.0003
0.0007
0.0211
-2.4331
4.4250
-1.0704
0.1449
9.3557
t-Statistic
6.6330*
-0.8608
1.8292
11.4827*
0.4500
-1.4989
2.6104*
-0.8251
0.0463
2.4707*
Pseudo-R 2
0.93
n = 88
Marginal impact
-0.1999
0.0000
0.0000
0.0014
-0.1714
0.3117
-0.0754
0.0102
0.6590
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Nicaragua
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
1542.6402
-2.8482
0.0003
0.0007
0.0443
-2.5793
4.5108
0.4493
-2.5347
4.4354
t-Statistic
6.6330*
-82.0482*
1.6891
11.5663*
0.9019
-1.5070
2.5863*
0.3186
-0.7856
1.0431
Pseudo-R 2
0.92
n = 88
Marginal impact
-0.2006
0.0000
0.0000
0.0031
-0.1817
0.3177
0.0316
-0.1785
0.3124
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Panama
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
18538.4510
-3.5692
0.0027
0.0000
-0.0282
0.8298
5.2890
3.9037
2.9847
-1.6964
t-Statistic
6.2845*
-54.4894*
4.8621*
0.7076
-0.6185
0.3414
2.9812*
2.4210*
0.7336
-0.3477
Pseudo-R 2
0.90
n=79
Marginal impact
-0.1583
0.0001
0.0000
-0.0012
0.0360
0.2346
0.1732
0.1324
-0.0752
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Panama
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
22730.2630
-3.5610
0.0025
0.0000
0.0169
3.8410
2.9267
3.7962
7.2748
0.0378
t-Statistic
6.2845*
-58.9989*
5.0049*
0.8689
0.4212
1.7993
1.7845
2.5088 *
2.0231*
0.0089
Pseudo-R 2
0.92
n=79
Marginal impact
-0.1580
0.0001
0.0000
0.0007
0.1704
0.1298
0.1684
0.3228
0.0016
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : El Salvador
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
12356.7010
-2.8443
-0.0009
-0.0002
0.0122
-0.2279
-4.8703
-5.2278
6.0138
5.3416
t-Statistic
8.8872*
-0.6085
-2.7607*
-6.3749*
0.2490
-0.1677
-2.098*
-4.7191*
2.0177*
2.1568*
Pseudo-R 2
0.84
n=158
Marginal impact
-0.1169
0.0000
0.0000
0.0005
-0.0093
-0.2001
-0.2148
0.2471
0.2195
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : El Salvador
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
12060.6110
-2.8065
-0.0010
-0.0002
-0.0042
-0.6994
-6.4686
-5.0776
4.1376
4.6936
t-Statistic
8.8872*
-56.9263*
-3.2045*
-5.9211*
-0.0873
-0.5013
-2.7779*
-4.5581*
1.1785
1.8724
Pseudo-R 2
0.84
n=158
Marginal impact
-0.1153
0.0000
0.0000
-0.0001
-0.0287
-0.2658
-0.2087
0.1700
0.1929
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Guatemala
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
98850.3980
-2.4850
-0.0003
0.0000
-0.0392
0.1827
0.0187
0.4703
1.4001
-2.7976
t-Statistic
6.5954*
-108.8519*
-1.3507
-7.1092*
-1.9851*
0.2728
0.0258
0.8557
0.9680
-1.6390
Pseudo-R 2
0.94
n=87
Marginal impact
-0.1706
0.0000
0.0000
-0.0026
0.0125
0.0012
0.0322
0.0961
-0.1920
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Guatemala
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
112215.4600
-2.4988
-0.0001
-0.0001
-0.0312
1.0493
0.3331
-4.5568
-0.1321
4.5915
t-Statistic
6.5954*
-114.0998*
-0.6858
-7.9309
-1.7057
1.6601
0.4917
-1.2849
-0.2452
3.4750*
Pseudo-R 2
0.95
n=87
Marginal impact
-0.1715
0.0000
0.0000
-0.0021
0.0720
0.0228
-0.4933
-0.0090
0.3152
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Honduras
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
42949.2410
-2.5086
-0.0022
0.0001
-0.0422
1.7350
-4.7390
-0.5684
6.9203
-8.2482
t-Statistic
6.9280*
-91.5870*
-6.9673*
2.6006*
-1.4747
1.6983
-4.0398 *
-0.7899
3.5472*
-3.1248*
Pseudo-R 2
0.89
n = 96
Marginal impact
-0.1488
-0.0001
0.0000
-0.0025
0.1029
-0.2811
-0.0337
0.4105
-0.4893
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Honduras
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
40786.4220
-2.53370
-0.0020
0.0000
-0.0717
1.4496
-3.6331
-0.9885
8.8814
-3.4222
t-Statistic
6.92800*
-83.76780*
-5.856*
1.8481
-2.4610*
1.3645
-3.0063*
-1.3048
3.4797*
-1.1378
Pseudo-R 2
0.82
n = 96
Marginal impact
-0.1503
-0.0001
0.0000
-0.0042
0.0859
-0.2155
-0.0586
0.5268
-0.2030
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Nicaragua
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
24887.6730
-2.5849
0.0000
0.0000
-0.0517
1.9110
-1.8924
-1.6111
-0.1888
-1.9268
t-Statistic
6.6331
-93.9232
-0.1936
1.7835
-1.3305
1.4371
-1.3269
-1.4993
-0.0705
-0.6037
Pseudo-R 2
0.93
n = 88
Marginal impact
-0.1819
0.0000
0.0000
-0.0036
0.1345
-0.1332
-0.1134
-0.0132
-0.1356
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Nicaragua
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
24455.4160
-2.5906
0.0000
0.0001
-0.0631
0.9053
-2.0921
-0.3777
-3.7453
1.5728
t-Statistic
6.6331
-0.9367
-0.2480
2.0711
-1.6324
0.6751
-1.4888
-0.3424
-1.4327
0.4624
Pseudo-R 2
0.92
n = 88
Marginal impact
-0.1823
0.0000
0.0000
-0.0044
0.0637
-0.1472
-0.0265
-0.2636
0.1107
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : Panama
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
36024.5710
-2.5208
0.0007
0.0000
0.0122
-2.6478
2.8865
-3.6265
-0.0136
-1.7403
t-Statistic
6.2848
-70.9880
2.421
-0.2658
0.4936
-2.0075
3.0019
-4.1435
-0.0061
-0.6536
Pseudo-R 2
0.78
n=79
Marginal impact
-0.2049
0.0000
0.0000
0.0009
-0.2153
0.2347
-0.2949
-0.0011
-0.1415
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : Panama
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
35514.6610
-2.4830
0.0003
0.0000
0.0297
-1.2170
3.6239
-4.1984
-1.3114
0.1467
t-Statistic
6.2848
-67.7143
1.2435
0.4565
1.2146
-0.9407
3.6752
-4.5776
-0.5881
0.0564
Pseudo-R 2
0.77
n=79
Marginal impact
-0.2019
0.0000
0.0000
0.0024
-0.0989
0.2947
-0.3414
-0.1066
0.0119
Model: YFt - Breads and Cereals,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
7878.1405
-3.3370
0.0001
0.0000
0.0927
2.9336
-3.1597
-1.7938
23.7910
-15.5140
t-Statistic
9.8974*
-113.6013*
0.7738
-2.2829*
1.7639
1.69909
-1.3018
-1.2593
6.4806*
-4.6612*
Pseudo-R 2
0.59
n=196
Marginal impact
-0.1163
0.0000
0.0000
0.0032
0.1023
-1.1019
-0.0625
0.8297
-0.5410
Model: YFt - Meat,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
6230.1340
-3.6794
0.0003
0.0000
0.0166
4.7924
-4.9458
-2.1258
2.0895
-19.0601
t-Statistic
9.8957
-99.4240*
1.4577
0.5838
0.2499
2.1591*
-1.6081
-1.1622
4.4363*
-4.3945*
Pseudo-R 2
0.45
n = 196
Marginal impact
-0.0988
0.0000
0.0000
0.0004
0.1287
-0.1328
-0.0571
0.5613
-0.5120
Model: YFt - Dairy Products and eggs,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
15087.3300
-3.4559
0.0004
0.000
0.0178
1.2827
-2.8169
3.5970
11.6713
-9.3770
t-Statistic
9.8983
-161.4307*
3.5551*
-1.6811
0.4656
0.9996
-1.6028
3.4580
4.3040*
-3.8050*
Pseudo-R 2
0.45
n=196
Marginal impact
-0.11708
0.0000
0.0000
0.0006
0.0434
-0.0954
0.1218
0.3954
-0.3176
Model: YFt - Other foods,
Xt = (EconAc
volsoy
Imp
volrice
roil
volcbot
volcorn
volkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
2630.1720
-2.9354
0.0000
0.0000
-0.1669
4.0297
-3.1935
-6.8022
21.3702
-13.4726
t-Statistic
9.8961
-70.3500*
0.0810
0.6173
-2.2319*
1.6014
-0.9182
-3.2785*
4.0313*
-2.8043*
Pseudo-R 2
0.42
n = 196
Marginal impact
-0.1505
0.0000
0.0000
-0.0085
0.2067
-0.1638
-0.3489
1.0962
-0.6911
Model: YFt - Breads and Cereals,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
8223.3444
-3.3688
0.0003
0.0000
0.1129
3.8016
-3.0465
-2.6718
25.3344
-14.6022
t-Statistic
9.8975
-108.1245*
1.8044
-3.3699*
2.1851243*
2.2541*
-1.2818
-1.9362*
6.8474*
-4.5111*
Pseudo-R 2
0.45
n=196
Marginal impact
-0.1174
0.0000
0.0000
0.0039
0.1325
-0.1062
-0.0931
0.8835
-0.5092
Model: YFt - Meat,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
6392.2242
-3.7041
0.0005
0.0000
0.0289
3.6510262
-5.7907
-3.7430
23.7709
-16.8039
t-Statistic
9.8958*
-93.5701*
2.2166*
-0.25107
0.4379
1.6557844
-1.9026
-2.0863*
4.9838*
-3.9679*
Pseudo-R 2
0.47
n=196
Marginal impact
-0.0995
0.0000
0.0000
0.0007.
0.0980
-0.1555
-1.0055
0.6385
-0.4514
Model: YFt - Dairy products and eggs,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
15898.1930
-3.4930
0.0007
0.0000
0.0337
1.0322
-3.2567
3.6454
15.3850
-9.1957
t-Statistic
9.8984*
-15.4867*
4.9618*
-3.017*
0.8973
0.8250
-1.9006
3.6323*
5.6644*
-3.8571*
Pseudo-R 2
0.49
n = 196
Marginal impact
-1.1834
0.0000
0.0000
0.0011
0.0349
-0.1103
0.1235
0.5212
-0.3115
Model: YFt - Other foods,
Xt = (EconAc
Lvolsoy
Imp
Lvolrice
roil
Lvolcbot
Lvolcorn
Lvolkcbt)
Table : India
Parameter
φ
θ0
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
Estimate
2676.8000
-2.9691
0.0001
0.0000
-0.1735
3.3521
-5.7748
-5.3801
25.2161
-13.2572
t-Statistic
9.8962*
-66.4044 *
0.6982
0.2103
-2.3303*
1.3424
-1.6695
-2.6484*
4.6849*
-2.8063*
Pseudo-R 2
0.43
n = 196
Marginal impact
-0.1523
0.0000
0.0000
-0.0089
0.1719
-0.29624
-0.2760
1.2935
-0.6800
Future research
I
We are working on the asymptotic properties of a semiparametric
estimator for the model
E (YtF |h1/2 (rt−1 , · · · , rt−L ), Wt ) = g −1 (θh1/2 (rt−1 , · · · , rt−L )+m(Wt ))
Implementation of this model and estimator will be hindered by the
small time series.
I
Different food groupings might give stronger results
I
Volatility sequence is stochastic and dependent!