# What Drives Inflation in India: Overheating or Input Costs?

Neeraj Hatekar, Ashutosh Sharma, Savita Kulkarni

This study takes a closer look at some of the drivers of inflation in manufacturing prices in India. It indicates that “overheating”, which has recently acquired policy focus, drives inflation in the short run, whereas international materials and energy prices drive inflation over the short as well as the long run. The study implies that restrictive monetary policy might be of only limited relevance in controlling non-food inflation. Public policy aimed at minimising the impact of input cost shocks might work better in the long run. In the meanwhile, the restrictive monetary policy followed by the Reserve Bank of India might worsen the downturn that has begun in January 2010.

The authors are grateful for helpful comments from an anonymous referee. However, the responsibility for errors that remain lies solely with the authors.

Neeraj Hatekar (neeraj.hatekar@gmail.com) is at the Centre for Computational Social Sciences and Department of Economics, University of Mumbai, Mumbai. Ashutosh Sharma (ashuofrabi@gmail.com) and Savita Kulkarni (kulkarni_sav@rediffmail.com) are PhD scholars, Department of Economics, University of Mumbai, Mumbai.

##### 1 Introduction

nﬂation is required to be kept under check for sustained economic growth, promotion of investment and for prevention of deterioration in the living standards of the masses. However, inﬂationary pressures have built up recently in the Emerging Market Economies (EMEs) including India. In this regard, the International Monetary Fund (IMF) had cited demand side explanation for the growing inﬂationary pressure in EMEs. The IMF’s World Economic Outlook (April 2011) attributes the current inﬂationary environment in EMEs to an “overheating” phenomenon in these economies. On the other hand, the current global inﬂationary pressure could also be attributed to supply-side factors such as rising input prices – mainly oil and metals. Higher commodity prices can cause both a reduction in real output and a cost push inﬂation. It is well established now that the oil price shocks in the 1970s were a major source of inﬂationary pressure in the global economy which challenged the famous traditional Phillips Curve. Given the global inﬂationary environment, the IMF’s argument of overheating could be one of the contemporaneous causes of rising inﬂationary pressure in a country like India, but it cannot be the exclusive explanation. The rising commodities prices in the international market can be equally responsible for exerting inﬂationary pressure in India which is one of the major importers of petroleum products. Therefore, the booms and busts in international commodity prices cannot be neglected while discussing the Indian domestic price level.Kaldor (1976) argued that any substantial increase in commodity prices will have a powerful inﬂationary effect on industrial costs and prices. He further argued that any large change in commodity prices will have a dampening effect on industrial activity, in addition to indirect price and wage effects. The traditional view for relying on commodity prices in predicting the general price level is that these prices usually enter with lags in the costs of output prices. Labys and Maizels (1993) by employing Granger causality (hereafter GC) tests between commodity prices and macroeconomic variables for major countries of the Organisation for Economic Cooperation and Development (OECD) conclude that the highest degree of causality exists between international primary commodity prices and selected macroeconomic indicators including national prices and industrial production. Labys and Maizels (1993) showed that commodity prices led changes in interest rates and, to a lesser extent, adjustments in the money supply, suggesting that the major countries did, in fact, practise monetary policy adjustments of some form, to counter major commodity price swings.

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Economic & Political WeeklySPECIAL ARTICLE

Very recently, in the monetary policy statement for 2011-12, the Reserve Bank of India (RBI) had also shown serious concern regarding inﬂationary risks emanating from rising commodity prices and its downside risks to growth. Considering the domestic demand-supply balance and the global trends in commodity prices, the RBI had revised the baseline projection for wholesale price index (WPI) inﬂation for March 2012 at 6% with an upward bias. RBI began exiting from the crisis-driven accommodative policy in October 2009 and since then had kept revising the policy rates upwards. However, in view of the overall inﬂation scenario, the RBI announced a further upward revision in the policy rates in its monetary policy statement for 2011-12.

The RBI concern over rising international commodity prices makes it imperative to look for how international commodity prices affect India’s domestic price level. Rakshit (2011) had discussed the linkages between selected macroeconomic indicators including domestic sectoral and overall price indices in the Indian context in the time-domain framework. However, it is reasonable to believe that domestic sectoral price indices act as a lagging indicator and international commodity prices as a leading one (mainly in the case of industrial inputs). Moreover, the strength of “pass-through” of international commodities prices in the domestic price level and industrial activity may vary over different time horizons. In other words, the strength of causality running from international commodities prices into domestic price level and output may vary in the short run, over the business cycle frequencies and in the long run. Therefore, the usual time domain representation may not necessarily be the most revealing, in cases such as business cycle analysis, because important economic information may be hidden in the frequency content of the time series. In order to access this information we need to transform a time series from the time domain and represent it in another domain. Granger (1969) had suggested that a spectral-density approach of GC would give a richer and more comprehensive picture than a one-shot GC measure that is supposed to apply across all periodicities.

In what follows, this essay aims to assess the “pass-through” of international commodities prices to manufacturing products prices and its effect on manufacturing sector output. We also analyse the IMF’s overheating viewpoint in the Indian context. We apply the GC test in the frequency domain proposed by Lemmens et al (2008) to see if the observed causality correspond to the business cycle frequencies or from too high (short run) or too low (long run) frequencies (see Appendix, p 50).1

The plan of the paper is as follows. Section 2 describes the data and Section 3 discusses the causality test methodology. Section 4 provides the empirical ﬁnding and Section 5 concludes the paper.

##### 2 The Data

For this study we have used monthly data for the period 1994:4 (fourth quarter of 1994) to 2010:12 (twelfth quarter of 2010). We have considered India’s Index of Industrial Production (IIP) for the manufacturing sector at the base year 1993-94 as a measure of output. WPI of manufacturing prices has been considered as a proxy for prices. The base years of the WPI series are 1993-94 and

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august 20, 2011 vol xlvi no 342004-05. We have extrapolated the new series for the period 1994:4 to 2004:3 by using the growth rate of the previous series with base year 1993-94. Data for IIP was collected from the Central Statistical Organisation (CSO – http://mospi.nic.in/cso_test1.htm) and data for the WPI from the Ofﬁce of the Economic Adviser (Government of India). Monthly (non-seasonally-adjusted) data for the Industrial Materials Index (IMI) and Energy Index (EI) at base year 2005 had been taken from the IMF’s primary commodity prices database (http://www.imf.org/external/np/res/commod/index.asp). In the IMI, agriculture raw materials have a nearly 42% weight and metals has 58%. In the EI, spot crude p etroleum has an 85% weight, natural gas has an 11% weight and coal has a 4% weight.

Figure 1 present the year-on-year (Y-O-Y) growth rate of the monthly IMI and EI and Figure 2 presents the Y-O-Y growth rate of India’s monthly WPI for the manufacturing sector for the period 2008:1 to 2010:12 which broadly incorporates the recent crises period. Figure 1 suggests that the growth rate of IMI and EI started falling from the middle of 2008 and then again shown an uptrend from the mid of the year 2009.

Figure 1: Year-on-Year Growth Rate of IMI and EI with Base Year 2005

Y-O-Y percentage growth

Source: IMF’s primary commodity prices database (http://www.imf.org/external/np/res/ commod/index.asp).

A similar trend was observed in India’s WPI for the manufacturing sector Y-O-Y growth rate (Figure 2). The WPI for the manufacturing sector Y-O-Y growth rate started falling after reaching 8.55% in August 2008 and remained negative for three consecutive months, 2009:6-2009:8. However, Y-O-Y growth of WPI for the manufacturing sector witnessed a reversal in its movement in September 2009. Prima facie, Y-O-Y growth rate movement in IMI and EI and somewhat a similar trend in India’s WPI for manufacturing provide evidence in support of the transmission of international commodities prices to the domestic price level.

Figure 2: Year-on-Year Growth Rate of India’s WPI for Manufacturing Sector at Base Year 2004-05

Y-O-Y percentage growth 10

-2

Source: Office of the Economic Adviser (http://eaindustry.nic.in/#), Government of India.

SPECIAL ARTICLE

Granger coefficient of coherence

be necessary to use seasonal rather than annual differencing to attain stationarity. Therefore, we have applied the Beaulieu and Miron methodology for testing unit roots in monthly data (Beaulieu and Miron 1993). They used the approach developed by Hylleberg et al (1990) to derive the mechanics of another pro

0.4

0.2

cedure for testing seasonal unit roots using monthly data. All the four variables, IIP, WPI, IMI and EI have been logarithmically transformed. The unit roots test results are presented

0.3 0.6 0.9 1.2 1.5 1.8 2.0 2.3 2.6 2.9 3.0

in Table 1. Frequency The dashed line represents the critical value for the null hypothesis, at the 5% level of significance.

Table 1: Beaulieu and Miron Test for Integration at Seasonal and Non-Seasonal Frequencies

In Figure 4 we have presented the result of the Granger causal

ʌ ʌ

ʌ ± ʌ

0

± ʌ ± ʌ

Granger coefficient ofcoherence

stationary in levels. Therefore we cannot reject the null hypothesis, i e, the presence of unit roots at seasonal frequencies as well as at zero frequency in all the four time series, IIP, WPI, IMI and EI. Following from the above result, all the four variables were season

0.2

ally differenced and further ﬁrst differenced, which leaves us

0

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.0 2.3 2.6 2.9 3.0

with 188 observations. The differenced series ¨ 12¨of IIP, WPI, IMI

and EI were found stationary.

Frequency

As in Figure 3.

IIP (Stat) | -1.05 | -3.89 | 4.61 | 4.65 | 4.93 | 11.91 | 8.14 |
---|---|---|---|---|---|---|---|

(P-value) | (0.1)* | (0.01) | (0.01) | (0.01) | (0.01) | (0.1)* | (0.09)* |

WPI (Stat) | -1.76 | -4.56 | 14.40 | 11.17 | 11.83 | 14.95 | 9.75 |

(P-value) | (0.1)* | (0.01) | (0.1)* | (0.1)* | (0.1)* | (0.1)* | (0.1)* |

IMI (Stat) | -1.60 | -5.06 | 12.35 | 6.25 | 17.50 | 16.59 | 6.82 |

(P-value) | (0.1)* | (0.01) | (0.1)* | (0.02) | (0.1)* | (0.1)* | (0.04) |

EI (Stat) | -3.00 | -3.73 | 23.15 | 24.33 | 20.75 | 15.30 | 5.68 |

(P-value) | (0.09)* | (0.01) | (0.1)* | (0.1)* | (0.1)* | (0.1)* | (0.01) |

In the parenthesis, associated P-value has been given. The (*) reflects the fact that we cannot reject the null hypothesis of presence of unit roots at 5% level of significance. The test has been performed using the software package “uroot” in ‘R’.

The results in Table 1 show that all the variables were nonity running from EI to WPI. At 5% level of signﬁcance, EI Granger causes WPI at frequencies corresponding to cycles longer than six months. Therfore, EI Granger causes WPI at frequencies corresponding to short as well as long run business cycle frequencies. The Granger coefﬁcient of coherence reaches its peak, i e, 0.37 at frequencies corresponding to long cycles, which highlights that the causality running from EI to domestic prices becomes much stronger at longer cycles.

Figure 4: Causality from International Energy Index to WPI Manufacturing

0.4

To apply the above mentioned GC test methodology, all the four differenced series (i e, ¨ 12¨of the series) of IIP, WPI, IMI and EI have been ﬁltered using ARMA models to obtain the innovation series. After ARMA ﬁltering the series and adjusting for lags, we are left with 176 observations. Therefore, we can consider N = 88 cycles of different frequencies, with the shortest possible cycle of two months and longest cycle of 176 months. We have used lag length M = ¥7 . The frequency (Ȝon the horizontal axis can be translated into a cycle or periodicity of T months by T = 2ʌȜ where T is the period.

The results presented in Figures 3 and 4 suggest that both IMI and EI Granger causes WPI in frequencies corresponding to short as well as long run business cycle frequencies. However, the causlity running from IMI to WPI is relatively stronger compared to that of from EI.

We now look for evidence for causality running from IMI and EI to IIP. In Figure 5 (p 49) we have presented the result of Granger causality running from IMI to IIP. The result indicates that at 5% level of signiﬁcance IMI Granger causes IIP at frequencies corresponding to 12-58 month cycles. This represents that

Analysing time series in the frequency-domain, i e, spectral analysis, could be helpful in supplementing the information obtained by time domain analysis (Granger 1969 and Priestley 1981). Spectral analysis highlights the cyclical properties of data. In particular, it allows us to decompose the variance of a stationary time series by variance attributed to cycles of various frequency. Similarly, the co-movement in a pair of stationary time series can be decomposed by cycles of various periodicities. This study uses the methodology proposed by Lemmens et al (2008) to decompose causality between pairs of stationary times series at cycles of different periodicities. The details of the methodology are given in Appendix 1.

##### 3 Empirical Findings

We ﬁrst test for stationarity of all the four series. Franses (1991) suggested that if the stochastic process is non-stationary, it is important that we investigate the presence of unit roots at all frequencies, not just the long run, because in this case it would

##### 3.1 Causality between International Commodity Prices, Domestic Prices and Output

We ﬁrst test the causality running from IMI to WPI manufacturing. The results presented in Figure 3 suggest that at 5% level of signiﬁcance IMI Granger causes WPI at higher as well as at lower frequencies. IMI Granger causes WPI manufacturing at frequencies corresponding to a three-month cycle and then from a 6-176 month cycle. We can say that IMI Granger causes WPI in the short run, at business cycle frequencies and also in the long run. The estimated coefﬁcient of coherence reaches its peak, i e, 0.45 at frequencies corresponding to a 176-month cycle. The results thus suggest that IMI can be considered as one of the leading indicators to predict WPI manufacturing in India. Therefore, rising international inputs prices do pass through the domestic price level and the transmission of international commodities prices in domestic price level strenghtens at longer cycles.

Figure 3: Causality from International Industrial Material Index to WPI Manufacturing

0.6

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SPECIAL ARTICLE

causality running from IMI to IIP is signiﬁcant in the short run and at the business cycle frequencies. The highest Granger coefﬁcient of coherence of 0.30 is associated with frequencies of a 16-month cycle, and at longer cycles the causality starts falling. The causality running from IMI to IIP is relatively weak compared with causality running from IMI to WPI. However, causality from IMI to IIP reﬂects that ﬂuctations in commodity prices also lead to output adjustments, although at a relatively longer cycles compared to prices.

###### Figure 5: Causality Running from Industrial Material Index to IIP

Granger coefficient of

coherence

0.4

0.2

0

Frequency

set equal to 1,29,600 following Ravn and Uhlig (2002) suggestion for monthly data. We have taken the recent 176 observations of the ﬂuctuations in the IIP series from the trend to examine the Granger causality running from output gap to WPI manufacturing sector, because as discussed above we have 176 observations of the WPI price series after ﬁltering and adjusting for lags.

The result for causality running from the output gap to WPI has been presented in Figure 7. At 5% level of signiﬁcance, ﬂuctuations in the IIP from the trend Granger cause WPI manufacturing at frequencies corresponding to two-three months cycle. Therefore, we argue that the output gap drives domestic inﬂation only in the short run and not at the business cycle frequencies or in the long run. The absence of causality running from the output gap to prices over the business cycle frequencies and in the long run indicates that supply side does

As in Figure 3.

catch up quickly. Figure 6 presents the results of the Granger coefﬁcient of Figure 7: Causality from Output Gap to WPI Manufacturing

coherence for causality from the EI to IIP manufacturing. In con

0.4

trast to the general proposition that higher crude oil prices add cost pressure on Indian industries, forcing them to cut output, we do not ﬁnd any causality between EI and IIP manufacturing. As shown in Figure 6, we do not ﬁnd evidence in favour of EI (statistically signiﬁcantly) “Granger causing” IIP at any frequen-

cies. One reason for this could be the cushioning of industrial

output through ﬁscal stimulus, such as higher subsidy. However,

0

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.0 2.3 2.6 2.9 3.0

Frequency

As in Figure 3.

the incomplete “pass-through” of higher crude prices through In this paper, the output gap has been measured by using the expansionary ﬁscal policy may cushion the industrial output but Hodrick-Prescott decomposition. However, the arbitrary choice of the smoothing parameter is a limitation of this method. In

###### Figure 6: Causality from International Energy Index to IIP Manufacturing

0.4

order to check the robustness of our results, we also used the

Beveridge-Nelson (B-N) decomposition (1981) approach to separate out the trend and cycle component of the index of industrial production. Our results show that the output gap measured using this approach does not “Granger cause” inﬂation at any frequency. Thus, our conclusion that there is only limited evidence

0

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.0 2.3 2.6 2.9 3.0

for the overheating hypothesis in the short run and none for the

Frequency

As in Figure 3.

long run is strengthened. it will undoubtedly worsen the ﬁscal burden of the government and may also add to inﬂationary pressure. 4 Conclusions

The salient feature of this study has been the frequency domain

3.2 Output Gap and Domestic Prices approach to uncover the causality relation between international We now try to assess the veracity of the IMF’s argument of over-commodity prices and domestic prices and output using monthly heating in the Indian context. In practice potential output is not data for the period 1994:4 to 2010:12. observed and has to be approximated. A

Figure 8: Stochastic Trend of IIP Manufacturing Y-O-Y Growth using B-N Decomposition Methodology (%)

widely used method for obtaining the poten | 25 | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

tial output and output gap is to apply the | |||||||||||||||||

Hodrick-Prescott ﬁlter (Hodrick and Prescott | 20 | ||||||||||||||||

1997) on actual output series. In an attempt to | |||||||||||||||||

investigate the applicability of IMF’s overheat | 15 | ||||||||||||||||

ing views, we have taken the ﬂuctuations | |||||||||||||||||

from the trend of actual output as a measure | 10 | ||||||||||||||||

of the output gap. We have applied the Hodrick | 5 | ||||||||||||||||

Prescott ﬁlter on the logarithmic transformed | |||||||||||||||||

output series (Manufacturing sector IIP) to ob | 0 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

tain the ﬂuctuations in the actual series from | 1995M06 | 1996M06 1997M06 1998M06 | 1999M06 2000M06 2001M06 2002M06 2003M06 | 2004M06 2005M06 | 2006M06 2007M06 2008M06 2009M06 2010M06 | ||||||||||||

the trend output. The Hodrick-Prescott Ȝ was | -5 |

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###### SPECIAL ARTICLE

The study ﬁnds that the IMF’s argument of overheating is Beveridge-Nelson) among the several potential approaches to at best relevant only in the short run in the Indian context. measuring the output gap. The results are clearly sensitive to However, the RBI’s concern about rising international commodity how the output gap is measured. Further work along these lines prices seems to be more relevant in driving domestic inﬂation is in progress. Having said this, our results so far do indicate that over the business cycle in the long run. We found that industrial an excessive concern with overheating may not be as successful material prices “Granger causes” both prices and output over the as envisaged. Restrictive monetary policy has an adverse business cycle frequency. Energy prices Granger cause only impact on growth and this becomes signiﬁcant in the context of domestic prices and not output. Therefore, the recent rise in glo-a possible downturn that has already occurred in year-to-year bal commodity prices is a major threat to domestic prices and (deseasonalised) growth rate of manufacturing IIP as brought output. Restrictive monetary policy attempting to cool the over-out by Figure 8 (p 49). heated economy might be of only limited relevance in controlling This graph shows the calculated stochastic trend in the yearnon-food inﬂation, and might take a serious toll on growth. But to-year deseasonalised growth rate of manufacturing IIP. After a there is a signiﬁcant role for public policy in minimising the im-rapid rise from January 2009 to December 2009, the stochastic pact of international commodity prices and costs in particular on trend has turned down, and continues to move down in a very domestic prices. However, as an important caveat, we would like rapid manner. This situation will not be helped by making investto point out that we have used only two (Hodrick-Prescott and ment costly by raising interest rates.

Note

We have followed the Burns and Mitchell (1946) deﬁnition of a business cycle: “…in duration business cycles vary from more than one year to 10 or 12 years…”.

References

Beaulieu, J J and J A Miron (1993): “Seasonal Unit Roots in Aggregate US Data”, Journal of Econometrics, 55, 305-28.

Beveridge, S and C R Nelson (1981): “A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the ‘Business Cycle’”, Journal of Monetary Economics, 7, 151-74.

Burns, A F and W C Mitchell (1946): “Measuring Business Cycles”, National Bureau of Economic Research, New York.

Diebold, F X (2001): Elements of Forecasting (2nd ed.) (Ohio: South-Western).

Franses, P H (1991): “Seasonality, Non-stationarity and the Forecasting of Monthly Time Series”, International Journal of Forecasting, 7(2), 199-208.

Granger, C W J (1969): “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods”, Econometrica, 37, 424-38.

Hamilton, James D (1994): Time Series Analysis (Princeton, NJ: Princeton University Press).

Hodrick, R and E Prescott (1997): “Post-War US Business Cycles: An Empirical Investigation”, Journal of Money, Credit and Banking, 29(1), 1-16.

Hylleberg, S, R F Engle, C W J Granger and B S Yoo (1990): “Seasonal Integration and Cointegration”, Journal of Econometrics, 44, 215-38.

Kaldor, N (1976): “Inﬂation and Recession in the World Economy”, Economic Journal, 86, pp 703-14.

Labys, W C and A Maizels (1993): “Commodity Price Fluctuations and Macroeconomic Adjustments in the Developed Economies”, Journal of Policy Modelling, 15(3), 335-52.

Lemmens, A, C Croux and M G Dekimpe (2008): “Measuring and Testing Granger Causality over the Spectrum: An Application to European Production Expectation Surveys”, International Journal of Forecasting, 24, 414-31.

Pierce, D A (1979): “R-Squared Measures for Time Series”, Journal of the American Statistical Association, 74, 901-10.

Priestley, M B (1981): Spectral Analysis and Time Series (London: London Academic Press).

Rakshit, M (2011): “Inﬂation and Relative Prices in India 2006-10: Some Analytical and Policy Issues”, Economic Political Weekly, Vol 46, No 16, pp 41-54.

Ravn, M O and H Uhlig (2002): “On Adjusting the

Hodrick-Prescott Filter for the Frequency of

Observations”, The Review of Economics and Sta

tistics, 84(2), 371-75. Reserve Bank of India (2011): “Monetary Policy State

ment 2011-12”, RBI.

Appendix 1

In our study, we follow the bivariate GC test over the spectrum proposed by Lemmens et al (2008). They have reconsidered the original framework proposed by Pierce (1979), and proposed a testing procedure for Pierce’s spectral GC measure. Let X and Y be two stationary time series of

tt

length T. The goal is to test whether X Granger

t

causes Y at a given frequency Ȝ. Pierce’s meas

t

ure for GC (Pierce 1979) in the frequencydomain is performed on the univariate innovations series, u and v, derived from ﬁltering the

tt

X and Y as univariate ARMA processes, i e,

tt

Ĭx(L)X = Cx + ĭx(L)u...(1)

tt

Ĭy(L)Y = Cy + ĭy(L)v ...(2)

tt

where Ĭx(L) and Ĭy(L) are autoregressive polynomials, Ĭx(L) and Ĭy(L) are moving average polynomials and Cx and Cy potential deterministic components. The obtained innovation series u and v, are the series of importance in

tt

the GC test proposed by Lemmens et al (2008). Let 6XȜ and 6YȜbe the spectral density functions, or spectra, of u and v at frequency

tt

Ȝ@ʌ, deﬁned by

6XȜ ȖXNH±LȜN ...(3)

ʌN ±

...(4)

6YȜ ȖYNH±LȜN

ʌN ±

where ȖXN= Cov (u) and ȖYN= Cov

t, ut–k

(v) represent the autocovariances of u and

t, vt–ktv at lag k. The idea of the spectral representa

t

tion is that each time series may be decomposed into a sum of uncorrelated components, each related to a particular frequency Ȝ. The spectrum can be interpreted as a decomposition of the series variance by frequency. The portion of variance of the series occurring between any

august 20, 2011

two frequencies is given by area under the spectrum between those two frequencies. In other words, the area under 6XȜ and, 6YȜ between any two frequencies Ȝ and ȜGȜ, gives the portion of variance of u and v respectively,

tt

due to cyclical components in the frequency band (ȜȜGȜ). The cross spectrum represents the cross covariogram of two series in frequency-domain. It allows determining the relationship between two time series as a function of frequency. Let S (Ȝ)

uv

be the cross spectrum between u and vseries.

tt

The cross spectrum is a complex number, deﬁned as,

(Ȝ) = XYȜ) L4XY (Ȝ)

6XY

...(5)

ȈȖXYNH±LȜN

ʌN ±

where XY Ȝ called cospectrum and 4XY (Ȝ) called quadrature spectrum are respectively, the real and imaginary parts of the cross-spectrum and L

. Here ȖYXN= Cov (XWYW±N) represents the cross-covariance of XW and YW at lag N. The cospectrum 4XY (Ȝ) between two series XW and YWat frequency Ȝ can be interpreted as the covariance between two series XW and YWthat is attributable to cycles with frequency Ȝ. The quadrature spectrum looks for evidence of out-of-phase cycles (Hamilton 1994: 274). The cross-spectrum can be estimated nonparametrically by,

0

A

6XYȜ ȈZNȖAXYNH±LȜN

...(6)# {}

ʌN ±0

with ˆȖXYN = WN the empirical cross

29X YW covariances, and with window weights ZN, for N ±00 Eq (6) is called the weighted covariance estimator, and the weights ZN are selected as the Bartlett weighting scheme, i e,

N

0. The constant M determines the maximum lag order considered. The spectra of Eq (3) and (4) are estimated in a similar way. This cross-spectrum allows us to compute the coefﬁcient of coherence h(Ȝ) deﬁned as,

uv

_6XYȜ_

KXYȜ ...(7)
¥6XȜ6YȜ

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Economic Political WeeklySPECIAL ARTICLE

Coherence can be interpreted as the absolute value of a frequency speciﬁc correlation coefﬁcient. The squared coefﬁcient of coherence has an interpretation similar to the R-square in a regression context. Coherence thus takes values between and . Lemmens et al (2008) have shown that, under the null hypothesis that h(Ȝ) = , the estimated squared coefﬁcient of

uv

coherence at frequency Ȝ, with Ȝʌ when appropriately rescaled, converges to a chisquared distribution with 2 degrees of freedom, denoted by F .

A

G

Q±KXYȜ oF

...(8) where oG stands for convergence in distribution, with Q 7Ȉ0 Z. The null hypothesis

N ±0N

h(Ȝ) = versus h(Ȝ) is then rejected if

uv uv

A

FD

KXYȜ!

...(9)

Q

with FD being the ±Į quantile of the chisquared distribution with 2 degrees of freedom. The coefﬁcient of coherence in Eq (7) gives a measure of the strength of the linear association between two time series, frequency by frequency, but does not provide any information on the direction of the relationship between two processes. Lemmens et al (2008) have decomposed the cross-spectrum Eq (5) into three parts:

(i) 6XY , the instantaneous relationship between XW and YW; (ii) 66X , the directional rela-

Y

tionship between YW and lagged values of XW; and (iii) 6YX, the directional relationship b etween XW and lagged values of YW, i e,

O6XY 6XY 6YX @

6XY

±

ȖXYȖXYNH±LȜNȖXYNH±LȜNʌ @

N ±N

...(10) The proposed spectral measure of GC is based on the key property that XW does not Granger cause YW if and only if ȖXYN for all N. The goal is to test the predictive content of XW relative YW to which is given by the second part of Eq (10), i e,

±

@6XYȜ ȖXYNH±LȜN ...(11)

ʌ N ±

The Granger coefﬁcient of coherence is then given by,

6 O

XY

K O

XY

...(12)6X O6YO

Therefore, in the absence of GC, KXY O for every Ȝ in @ʌ. The Granger coefﬁcient of coherence takes values between zero and one, Pierce (1979). Granger coefﬁcient of coherence at frequency Ȝ is estimated by

6XY O

KXY O

6X O6Y O ...(13)

with 6XY O as in Eq (6), but with all weights ZN for N. The distribution of the estimator of the Granger coefﬁcient of coherence is derived from the distribution of the coefﬁcient of coherence Eq (8). Under the null hypothesis

^

h KXY O , the distribution of the squared estimated Granger coefﬁcient of coherence at frequency Ȝ, with Ȝʌ is given by,

G

Q±KÖ O oF ...(14)

XY

where Q is now replaced by Q7 ¦ Z .

N

N 0

Since the ZN s, with a positive index N, are set equal to zero when computing 6AXY O, in effect only the ZN with negative indices are taken into account. The null hypothesis KAXY O versus KAXY O is then rejected if

D

KXY O!

Q

...(15) We then compute the Granger coefﬁcient of coherence given be Eq (13) and test the signiﬁcance of causality by making use of Eq (15).

EPW Research Foundation (A UNIT OF SAMEEKSHA TRUST)

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