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BUS5ATE Advanced Time Series Analysis

Part 1: Implications of Unit Root

Asset Prices: the efficient market hypothesis shows that returns follow a stochastic, rather

unpredictable pattern and at such asset prices tend to exhibit a stochastic process or

presence of unit root. Evidently, asset prices could be risk-free in the case of fixed income

securities and mortgage-backed securities and risky in the case of equities, foreign exchange

and commodities. A typical case of unpredictability and stochastic movements can be

aligned to equity progression or trend analysis that follows a random walk both with drift

and without drift. In terms of time horizon, the prices of an asset is more likely to change in

the long-term than in the short-terms with much greater risk expected to affect the value

and return of the asset in the long-term. Previous empirical findings exemplify this axiom to

mean that indeed asset prices have a unit root - in other words they behave sporadically. A

typical instance would be the actions of the Central bank in deciding what path the

economy will take by altering the policy lending rate to either contract or expand the

monetary system, their consequential effect will transcend to volatility in asset prices

following a shift in investor’s perception about the expected rate of inflation and return on

investment (ROI).

Purchasing Power Parity Hypothesis: exchange rate is a core determinant in income

determination, for every good produced, same price is expected irrespective of the country

where the good is sold or purchased. The implication of the hypothesis is rather flawed

because trade policies differ by country which will affect cost of imports and exports both

ways, in addition currency value especially in the domestic market differs relatively,

empirically studies have rejected the null hypothesis of a unit root for real exchange rate

even though nominal exchange rate is supposed to match the national price levels.

Growth and Convergence: the periods between economic fluctuation and growth are

sensitive ones. Empirical evidence have increasingly shown that capital converge at same

level so if the zero mean means that the difference in per capita income between two

countries is transitory and fluctuates around zero. This implies that two countries per capita

output converges and becomes same in the absence of unit root.

Convergence of Real Interest Rate: the implication of having two nations with same interest

rate is predicated on international macroeconomics. Interest rate has been found to have

unit root which essentially means that capital markets of the two nations are not fully

integrated. In essence, a myriad of factors would definitely cause the interest rate to

dissipate overtime namely the dynamics of inflation rate, the likelihood for interest to

change overtime due to the actions of the respective countries Central bank, the intensity of

the financial market especially currencies and equities where offshore investments flow and

non-economic factors.

Part 2: Unit Root in Real Interest Rate

The consumption-based asset pricing model explains the rudiments of real interest rate

which is simply nominal interest rate less of inflation rate or simply put interest rate

adjusted for expected inflation rate. This defines real interest rate as core variable in the

determination and evaluation of policies at the Central bank (Taylor, 1993) and the neo-

classical growth model (Cass, 1965; Koopmans, 1965). The postulation was further

expressed with empirical results using U.S. data which summarises the persistence and

exploration of the cause of real interest rate.

The relation exhibits a persistent and substantial shock following the advanced periods of

post-war. A combination of unit root and cointegration test was found to be central in

analysing the effects of shocks on real interest rate in other words does real interest rate

behaves like a random walk. Further research also shows the non-linearity in variable

behaviour and the concept of fractional integration, both of which are linked to persistent

shocks and structural breaks reaction in between periods. For instance, pre-war and post-

war effects of shocks on interest rate are to be mentioned. However, a major limitation

emanates from the inability of the study to address the root causes of real interest rate.

Hence, the need for a theoretical model as opined by Lucas (1978), and Breeden (1979)

termed the consumption-based asset pricing model which discusses the components of real

interest rate at different time periods, taking into consideration the risk factor and the per

capita consumption. The U.S. data was used by Rose (1988) to empirically test the

theoretical model using the 3-month real treasury bills rate and annualised growth. It was

found that a huge divergence exist in the 1970s, 1980s and 2001-2005.

More elaborate theoretical model was employed to examine the changes in macroeconomic

policies majorly monetary and fiscal policy – with intent of altering the steady-state real

interest rate using the Euler’s equation. In summary most researchers were open to the idea

of allowing changes in preferences over the extended period (Clark, 2007).

The concept of forecasting the behaviour of inflation was also brought to light to ascertain

the level of real interest rate for all the maturity spectrum of U.S securities. The theoretical

postulation was also empirically tested using the concepts of ex-ante real interest rate

(EARR) and ex-post real interest rate (EPRR); since business agents and investors alike make

futuristic decision on the basis of expected inflation rate - in other words EARR is a measure

for examining decisions; the study also starts by drawing an analogy and distinction

between both concepts via time horizon and persistence properties (Sun and Philip, 2004).

The structural vector autoregression (SVAR) became the method of estimation after

evaluating unit root test in real interest rate notably the result yielded a non-rejection of the

null hypothesis of unit root in a study that also includes the U.S. 3month treasury bills,

inflation and EPRR. In summary, an extension was made including the updated results on

unit root and cointegration for U.S. data both of which presented mixed outcomes and

confidence interval for the sum of the AR coefficients.

Part 3: Basic Data Analysis

Here, we estimate the time plots and SACF of the variables 3-Month Treasury Constant

Maturity Rate, PCE deflator inflation rate, Ex post real interest rate (EPRR) and Consumption

growth rate.

The presence of serial correlation will mean that the SACF at all lags should be near zero

(significantly different from zero or not zero) and the Ljung-box Q-stat is insignificant.

However, the results presented below shows persistent and consistent residuals. In terms of

degree of persistence, ex post real interest rate (EPRR) and Consumption growth rate (CGR)

are likely to be forecastable given the persistent pattern of their time plots shown below

while 3-Month Treasury rate and PCE deflator inflation rate are likely to be forecastable

given the non-persistent or stochastic or volatile pattern of their time plots.

Fractional Integration (ARFIMA) as a test for degree of dependence: stationary processes

are more likely to have long memory in the presence of autocorrelation, which tends to

decay more slowly (Granger and Joyeux, 1980; Hosking, 1981).

Dependent Variable: DLOG(CGR) Method: Least Squares Date: 06/09/17 Time: 00:14 Sample (adjusted): 3 219 Included observations: 217 after adjustments Convergence achieved after 13 iterations MA Backcast: 2 Variable Coefficient Std. Error t-Statistic Prob. C 0.006381 0.008713 0.732341 0.4648 PCE 9.36E-05 0.002800 0.033427 0.9734 EPRR 0.000894 2.54E-05 35.20569 0.0000 _3TCM -0.001816 0.002714 -0.669117 0.5042 AR(1) 0.198698 0.070268 2.827727 0.0051 MA(1) -0.999822 0.028257 -35.38283 0.0000 R-squared 0.581020 Mean dependent var 0.038775 Adjusted R-squared 0.571091 S.D. dependent var 1.245687 S.E. of regression 0.815815 Akaike info criterion 2.458001 Sum squared resid 140.4318 Schwarz criterion 2.551455 Log likelihood -260.6932 Hannan-Quinn criter. 2.495753 F-statistic 58.52071 Durbin-Watson stat 2.063847 Prob(F-statistic) 0.000000 Inverted AR Roots .20 Inverted MA Roots 1.00

Correlogram of CGR

Date: 06/08/17 Time: 23:53

Sample: 1 219

Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 0.000 0.000 5.E-06 0.998

.|. | .|. | 2 -0.000 -0.000 5.E-06 1.000

.|. | .|. | 3 -0.000 -0.000 9.E-06 1.000

.|. | .|. | 4 0.000 0.000 9.E-06 1.000

.|. | .|. | 5 -0.000 -0.000 9.E-06 1.000

.|. | .|. | 6 -0.000 -0.000 1.E-05 1.000

.|. | .|. | 7 -0.000 -0.000 2.E-05 1.000

.|. | .|. | 8 -0.000 -0.000 3.E-05 1.000

Correlogram of EARR

Date: 06/08/17 Time: 23:55

Sample: 1 219

Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 -0.000 -0.000 4.E-06 0.998

.|. | .|. | 2 0.000 0.000 5.E-06 1.000

.|. | .|. | 3 -0.000 -0.000 5.E-06 1.000

.|. | .|. | 4 -0.000 -0.000 6.E-05 1.000

.|. | .|. | 5 -0.000 -0.000 6.E-05 1.000

.|. | .|. | 6 0.000 0.000 6.E-05 1.000

.|. | .|. | 7 -0.000 -0.000 7.E-05 1.000

.|. | .|. | 8 -0.000 -0.000 7.E-05 1.000 Correlogram of PCE

Date: 06/08/17 Time: 23:55

Sample: 1 219

Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|******| .|******| 1 0.846 0.846 158.75 0.000

.|******| .|** | 2 0.792 0.271 298.73 0.000

.|******| .|** | 3 0.779 0.234 434.76 0.000

.|***** | .|. | 4 0.727 -0.009 553.68 0.000

.|***** | *|. | 5 0.660 -0.099 652.21 0.000

.|***** | .|* | 6 0.644 0.080 746.53 0.000

.|**** | *|. | 7 0.584 -0.100 824.31 0.000

.|**** | *|. | 8 0.511 -0.112 884.22 0.000 Correlogram of _3TCM

Date: 06/08/17 Time: 23:56

Sample: 1 219

Included observations: 219 Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|******* .|******* 1 0.929 0.929 191.52 0.000

.|******| .|* | 2 0.877 0.104 363.05 0.000

.|******| .|** | 3 0.856 0.218 527.27 0.000

.|******| **|. | 4 0.797 -0.237 670.28 0.000

.|***** | .|. | 5 0.745 0.008 795.71 0.000

.|***** | *|. | 6 0.693 -0.130 904.86 0.000

.|***** | *|. | 7 0.626 -0.086 994.33 0.000

.|**** | .|* | 8 0.593 0.178 1074.9 0.000

TIME PLOTS OF THE VARIABLES

-12,000

-10,000

-8,000

-6,000

-4,000

-2,000

0

2,000

25 50 75 100 125 150 175 200

CGR

-12,000

-10,000

-8,000

-6,000

-4,000

-2,000

0

2,000

25 50 75 100 125 150 175 200

EPRR

-2

0

2

4

6

8

10

12

25 50 75 100 125 150 175 200

PCE

0

2

4

6

8

10

12

14

16

25 50 75 100 125 150 175 200

3TCM

Part 4: Replication

Table 4a: Unit Root Test Statistics, U.S. data, 1953:Q1-2007:Q3

The order of integration or level of difference used in the model forms the third column in

the table above where it is indicated that the 3-Month Treasury bills rate and Per capita

consumption growth are non-stationary at all levels.

Table 4b: Cointegration Test Statistics, U.S. 3-Month Treasury bill Rate, Inflation Rate,

EPRR and Per capita consumption growth (1953:Q1-2007:Q3)

Cointegration Test Variable (Dynamic OLS) Prob.* Significance

3-Month Treasury bill rate 0.0006 Yes

PCE deflator 0.0006 Yes

Ex post real interest rate 0.1304 No

Per capita consumption growth 0.0140 Yes

The 10 percent, 5 percent, and 1 percent critical values for the probability value of the

variables shows significance (p<0.05) except EPRR. The lag order for the cointegration model

used to compute the test statistic is 14 based on Schwarz criterion.

Summary of the Results and Discuss their Implications:

Table 4a and 4b provides a clear evidence that EPRR and 3-Month Treasury bill rate are

relatively not stochastic when compared to PCE deflator and Per capita consumption

growth. This relation is equally depicted in the time plots for the respective variables. To

test the hypothesis of no cointegration, we interpret the probability value from the Engle-

Granger cointegration- the equation rejects the null hypothesis of no cointegration at the

1% and 5% level. Table 4b also presents the dynamic ordinary least squares (OLS) which also

shows that the variables are significantly different from zero except EPRR

Variable ADF Order of Integration Level of significance

3-Month Treasury bill rate 0.886369 Non-stationary at all levels 1% & 5%

PCE deflator -15.5305 Stationary at 1st Difference Nil Ex post real interest rate -2.85439 Non-stationary at all levels Nil

Per capita consumption growth -15.5305 Stationary at 1st Difference 1% & 5%

A variety of economic models have been employed in the past to analyse the persistence of

real interest rate and how other variables play a part in the determination of interest rate

effects. Many empirical evidence have shown that real interest rate contain a unit root

A structural analysis might be important in revealing the cause of the persistent shock and

dependence. looking at the variables; 3-month treasury bills rate, consumption growth rate

,ex post real interest rate and personal consumption, it can be deciphered that all of the

variables respond to shock as shown in the fractional integration output. Hence, a test for

unit roots to further examine the past behaviour of real interest rates and its determinants.

Undoubtedly, inflation (personal consumption expenditure deflator) and per capita

consumption growth were stationary at order (I).

REFERENCES

Breeden, Douglas T. “An Intertemporal Asset Pricing Model with Stochastic Consumption and

Investment Opportunities.” Journal of Financial Economics, September 1979, 7(3), pp. 265-

96.

Cass, David. “Optimum Growth in an Aggregate Model of Capital Accumulation.” Review of

Economic Studies, July 1965, 32(3), pp. 233-40.

Clark, Gregory. A Farewell to Alms: A Brief History of the Economic World. Princeton, NJ: Princeton

University Press, 2007.

Granger, Clive W.J. and Joyeux, Roselyne. “An Introduction to Long-Memory Time Series Models and

Fractional Differencing.” Journal of Time Series Analysis, 1980, 1, pp. 15-39.

Hosking, J.R.M. “Fractional Differencing.” Biometrika, April 1981, 68(1), pp. 165-76.

Koopmans, Tjalling C. “On the Concept of Optimal Economic Growth,” in the Economic Approach to

Development Planning. Amsterdam: Elsevier, 1965, pp. 225-300.

Lucas, Robert E. “Asset Prices in an Exchange Economy.” Econometrica, November 1978, 46(6), pp.

1429-45.

Rose, Andrew K. “Is the Real Interest Rate Stable?” Journal of Finance, December 1988, 43(5), pp.

1095-112.

Sun, Yixiao and Phillips, Peter C.B. “Understanding the Fisher Equation.” Journal of Applied

Econometrics, November-December 2004, 19(7), pp. 869-86.

Taylor, John B. “Discretion versus Policy Rules in Practice.” Carnegie-Rochester Conference Series on

Public Policy, December 1993, 39, pp. 195-214.

APPENDIX

Fractional Integration

Dependent Variable: DLOG(CGR)

Method: Least Squares

Date: 06/09/17 Time: 00:14

Sample (adjusted): 3 219

Included observations: 217 after adjustments

Convergence achieved after 13 iterations

MA Backcast: 2 Variable Coefficient Std. Error t-Statistic Prob. C 0.006381 0.008713 0.732341 0.4648

PCE 9.36E-05 0.002800 0.033427 0.9734

EPRR 0.000894 2.54E-05 35.20569 0.0000

_3TCM -0.001816 0.002714 -0.669117 0.5042

AR(1) 0.198698 0.070268 2.827727 0.0051

MA(1) -0.999822 0.028257 -35.38283 0.0000 R-squared 0.581020 Mean dependent var 0.038775

Adjusted R-squared 0.571091 S.D. dependent var 1.245687

S.E. of regression 0.815815 Akaike info criterion 2.458001

Sum squared resid 140.4318 Schwarz criterion 2.551455

Log likelihood -260.6932 Hannan-Quinn criter. 2.495753

F-statistic 58.52071 Durbin-Watson stat 2.063847

Prob(F-statistic) 0.000000 Inverted AR Roots .20

Inverted MA Roots 1.00

Null Hypothesis: CGR has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic 0.886369 0.9952

Test critical values: 1% level -3.460313

5% level -2.874617

10% level -2.573817 *MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(CGR)


Date: 06/09/17 Time: 01:16


Included observations: 218 after adjustments Variable Coefficient Std. Error t-Statistic Prob.

CGR(-1) 22.88248 25.81598 0.886369 0.3764

C -94.17414 71.24199 -1.321891 0.1876 R-squared 0.003624 Mean dependent var -45.87443

Adjusted R-squared -0.000989 S.D. dependent var 677.2536


Sum squared resid 99171204 Schwarz criterion 15.91514



Prob(F-statistic) 0.376404

Null Hypothesis: D(CGR) has a unit root

Exogenous: Constant



5% level -2.874679



Dependent Variable: D(CGR,2)


Date: 06/09/17 Time: 01:16


Included observations: 217 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(CGR(-1)) 16.59933 22.58757 0.734888 0.4632








Null Hypothesis: D(CGR,2) has a unit root

Exogenous: Constant



5% level -2.874741



Dependent Variable: D(CGR,3)


Date: 06/09/17 Time: 01:16


Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(CGR(-1),2) 2.259183 13.63943 0.165636 0.8686








Null Hypothesis: PCE has a unit root

Exogenous: Constant

Lag Length: 2 (Automatic - based on SIC, maxlag=14) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.386041 0.1469


5% level -2.874741



Dependent Variable: D(PCE)


Date: 06/09/17 Time: 01:19


Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. PCE(-1) -0.084611 0.035461 -2.386041 0.0179

D(PCE(-1)) -0.362200 0.068943 -5.253631 0.0000

D(PCE(-2)) -0.227295 0.066559 -3.414915 0.0008

C 0.298905 0.146956 2.033982 0.0432 R-squared 0.189227 Mean dependent var -0.002495







Null Hypothesis: D(PCE) has a unit root

Exogenous: Constant



5% level -2.874741



Dependent Variable: D(PCE,2)


Date: 06/09/17 Time: 01:20


Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(PCE(-1)) -1.669831 0.107519 -15.53051 0.0000

D(PCE(-1),2) 0.256747 0.066121 3.882970 0.0001








Null Hypothesis: EPRR has a unit root

Exogenous: Constant



5% level -2.874617



Dependent Variable: D(EPRR)


Date: 06/09/17 Time: 01:21


Included observations: 218 after adjustments Variable Coefficient Std. Error t-Statistic Prob.

EPRR(-1) -11.96557 20.81428 -0.574873 0.5660








Null Hypothesis: D(EPRR,2) has a unit root

Exogenous: Constant



5% level -2.874804



Dependent Variable: D(EPRR,3)


Date: 06/09/17 Time: 01:22


Included observations: 215 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(EPRR(-1),2) -111.9242 39.21128 -2.854388 0.0047

D(EPRR(-1),3) 56.97723 21.94793 2.596018 0.0101








Null Hypothesis: D(PCE) has a unit root

Exogenous: Constant



5% level -2.874741



Dependent Variable: D(PCE,2)


Date: 06/09/17 Time: 01:25


Included observations: 216 after adjustments Variable Coefficient Std. Error t-Statistic Prob. D(PCE(-1)) -1.669831 0.107519 -15.53051 0.0000

D(PCE(-1),2) 0.256747 0.066121 3.882970 0.0001








Date: 06/09/17 Time: 01:46

Series: CGR EPRR _3TCM PCE

Sample: 1 219

Included observations: 219

Null hypothesis: Series are not cointegrated

Cointegrating equation deterministics: C

Automatic lags specification based on Schwarz criterion (maxlag=14)

Dependent tau-statistic Prob.* z-statistic Prob.*

CGR -4.260599 0.0377 -52.07718 0.0006

EPRR -4.262521 0.0375 -52.09807 0.0006

_3TCM -3.498753 0.1990 -25.32218 0.1304

PCE -4.129776 0.0524 -37.67988 0.0140 *MacKinnon (1996) p-values.

Intermediate Results:

CGR EPRR _3TCM PCE

Rho - 1 -0.481032 -0.481192 -0.261841 -0.299493

Rho S.E. 0.112902 0.112889 0.074838 0.072520

Residual variance 4.404948 4.404056 1.910262 2.007643

Long-run residual variance 1.137966 1.137889 0.386494 0.681122

Number of lags 5 5 3 2

Number of observations 213 213 215 216

Number of stochastic trends** 4 4 4 4 **Number of stochastic trends in asymptotic distribution

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