Page 121 - Invited Paper Session (IPS) - Volume 2

IPS 188 G. P. Samanta
2.3 Change in Indices/Variables, Inflation Rates and Codes/Symbols
Used for Different Variables
The k-period inflation rates for t-th month, based on CPIs are the log-
return or continuously compounding, are computed as

= 100 ∗ [ log − log − ] ….3

Where is the m-period change or inflation rate at t-th month; m=1 for

monthly change/inflation and m=12 for annual inflation/change; X=CPI-C,
CPI-U, CPI-R or Google trend indices GMPrice, GMInfl.
For convenience in referencing any transformation or derived variable for
a time series, say X, we used codes/symbols as follows; gX = annual
percentage changes as in Eqn. (3); lnX = log (); ∆X and ∆2X represent 1 and
st

nd
2 difference of X, respectively; eX and e2X are residual obtained by fitting
linear and quadratic time trends on X respectively; where X = CPI-C, CPI-U,
GMPrice, GMInfl or corresponding logtransformations.

3. Results
The assessment of the information content of a given Google search Index
is carried out in a multi-step process. First, we examine basic time series
properties, such as stationarity or non-stationarity, of offline price indices,
Google trend index, change in indices and inflation rates.
3.1 Tests for Unit-Root – Difference-Stationary and Trend-Stationary
Processes
We applied Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests
for examining if (a) log-transformed indices are stationary, and (b) if any
detected non-stationary series belongs to trend-stationary (TS) or difference-
stationary (DS) class. The equation considered for implementing ADF tests for
a time series Xt is of the following general form.
∆Xt = α + βt + ρXt−1 + ∑ 1 =1 ∆ − +

……(4)
Where ∆ is the difference operator; α, β, ρ and are unknown constants, and

is the usual error series.

In Eqn. (4), parameters of interest are β and ρ. If β=0 and ρ<1 then Xt is a
stationary, i.e. I(0) series, and if (β,ρ) =(0,1) then Xt has unit-root and belongs
to difference-stationary process, i.e. Xt is non-stationary I(1) and ∆Xt is I(0)
process. However, if β ≠ 0 and ρ < 1, then also Xt is non-stationary and belongs
to trend-stationary (TS) series. Removal of deterministic time trend from a TS
series would yield a stationary or I(0) series. The unit-root tests were first
carried out for for log(CPI-C), log(CPI-U), log(GMPrice) and log(GMInfl)
directly, and found some mixed results (Table 1). It appears that while lnCPI-C
and lnCPI-U are I(2) processes, lnGPPrice and lnGMInfl are I(1) processes.
Further, we examined the stationarity or unit-root properties of annual
inflation rates based on CPI-C and CPI-U, and the annual percentage change

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