# Predicting Stock Returns and Rate of Return Forecasting Models Project

Instructions:•

You must upload the work files from R (.prg). One for Part 1 and one for Part 2. Excel is

not suitable for this project, and it will not be accepted.

This project consists of two parts:

•

•

Part 1: Predicting Stock Returns.

Part 2: Forecasting models for the rate of inflation.

Part 1: Predicting Stock Returns.

Data Description:

Documentation for Stock_Returns_1931_2002

This file contains 2 monthly data series over the 1931:1-2002:12 sample period.

•

•

ExReturn: Excess Returns

ln_DivYield: 100×ln(dividend yield). (Multiplication by 100 means the changes are

interpreted as percentage points).

The data were supplied by Professor Motohiro Yogo of the University of Pennsylvania and were used in

his paper with John Campbell:

•

“Efficient Tests of Stock Return Predictability,” Journal of Financial Economics, 2006.

(Double click in the window below to access the data)

DATE_M

1931M01

1931M02

1931M03

1931M04

1931M05

1931M06

1931M07

1931M08

1931M09

EXRETURN

LN_DIVYIELD

5.9649584

-282.2329

10.3053054

-293.2089

-6.8408314

-287.8614

-10.4480653

-278.2477

-14.3580770

-265.4742

12.8502610

-280.5102

-6.6559179

-275.5950

0.0461485

-278.4424

-34.2583894

-247.1829

Some Background

exreturn: is the excess return on a broad-based index of stock prices, called the CRSP value-weighted

index, using monthly data from 1960:M1 to 2002:M12, where “M1” denotes the first month of the year

(January), “M2” denotes the second month, and so forth.

•

The monthly excess return is what you earn, in percentage terms, by purchasing a stock at the end

of the previous month and selling it at the end of this month, minus what you would have earned

had you purchased a safe asset (a U.S. Treasury bill). The return on the stock includes the capital

gain (or loss) from the change in price plus any dividends you receive during the month.

Calculating k-period stock returns:

One-period holding return:

𝑟𝑡,𝑡+1 = ln (

𝑟𝑡+1,𝑡+2 = ln (

𝑃𝑡+1

) = ln(𝑃𝑡+1 ) − ln(𝑃𝑡 ) = 𝑝𝑡+1 − 𝑝𝑡 = Δ𝑝𝑡+1

𝑃𝑡

𝑃𝑡+2

) = ln(𝑃𝑡+2 ) − ln(𝑃𝑡+1 ) = 𝑝𝑡+2 − 𝑝𝑡+1 = Δ𝑝𝑡+2

𝑃𝑡+1

Two-period holding return:

𝑃𝑡+2

𝑟𝑡,𝑡+2 = ln (

) = ln(𝑃𝑡+2 ) − ln(𝑃𝑡 ) = 𝑝𝑡+2 − 𝑝𝑡 = 𝑝𝑡+2 − 𝑝𝑡+1 + 𝑝𝑡+1 − 𝑝𝑡

𝑃𝑡

= Δ𝑝𝑡+2 + Δ𝑝𝑡+1 = 𝑟𝑡+1,𝑡+2 + 𝑟𝑡,𝑡+1

Other way

𝑟𝑡,𝑡+2 = ln (

𝑃𝑡+2

𝑃𝑡+2 𝑃𝑡+1

𝑃𝑡+1 𝑃𝑡+2

𝑃𝑡+1

𝑃𝑡+2

) = ln (

) = ln (

) = ln (

) + ln (

)

𝑃𝑡

𝑃𝑡 𝑃𝑡+1

𝑃𝑡 𝑃𝑡+1

𝑃𝑡

𝑃𝑡+1

= 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2

Three-period’s returns:

𝑟𝑡,𝑡+3 = ln (

𝑟𝑡,𝑡+3 = ln (

𝑃𝑡+3

) = ln(𝑃𝑡+3 ) − ln(𝑃𝑡 ) = 𝑝𝑡+3 − 𝑝𝑡

𝑃𝑡

𝑃𝑡+3

𝑃𝑡+3 𝑃𝑡+2 𝑃𝑡+1

𝑃𝑡+1 𝑃𝑡+2 𝑃𝑡+3

) = ln (

) = ln (

)

𝑃𝑡

𝑃𝑡 𝑃𝑡+2 𝑃𝑡+1

𝑃𝑡 𝑃𝑡+1 𝑃𝑡+2

𝑃𝑡+1

𝑃𝑡+2

𝑃𝑡+3

= ln (

) + ln (

) + ln (

) = 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2 + 𝑟𝑡+2,𝑡+3

𝑃𝑡

𝑃𝑡+1

𝑃𝑡+2

k-period’s returns:

𝑟𝑡,𝑡+𝑘 = ln (

𝑟𝑡,𝑡+𝑘 = ln (

𝑃𝑡+𝑘

) = ln(𝑃𝑡+𝑘 ) − ln(𝑃𝑡 ) = 𝑝𝑡+𝑘 − 𝑝𝑡

𝑃𝑡

𝑃𝑡+𝑘

𝑃𝑡+𝑘 𝑃𝑡+𝑘−1

𝑃𝑡+2 𝑃𝑡+1

𝑃𝑡+1 𝑃𝑡+2 𝑃𝑡+3

𝑃𝑡+𝑘

) = ln (

⋅ …⋅

) = ln (

⋅ …⋅

)

𝑃𝑡

𝑃𝑡 𝑃𝑡+𝑘−1

𝑃𝑡+2 𝑃𝑡+1

𝑃𝑡 𝑃𝑡+1 𝑃𝑡+2

𝑃𝑡+𝑘−1

𝑃𝑡+1

𝑃𝑡+2

𝑃𝑡+3

𝑃𝑡+𝑘

= ln (

) + ln (

) + ln (

) + ⋯ + ln (

)

𝑃𝑡

𝑃𝑡+1

𝑃𝑡+2

𝑃𝑡+𝑘−1

= 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2 + 𝑟𝑡+2,𝑡+3 + ⋯ + 𝑟𝑡+𝑘−1,𝑡+𝑘

When to apply a “buy and hold” strategy:

•

•

•

•

•

If you have a reliable “forecast” of future stock returns then an active “buy and hold” strategy

will make you rich quickly by beating the stock market.

If you think that the stock market will be going up, you should buy stocks today and sell them

later, before the market turns down. Forecasts based on past values of stock returns are sometimes

called “momentum” forecasts: If the value of a stock rose this month, perhaps it has momentum

and will also rise next month.

If so, then returns will be autocorrelated, and the autoregressive model will provide useful

forecasts. You can implement a momentum-based strategy for a specific stock or for a stock

index that measures the overall value of the market.

From another point of view, we can use autoregressive models to test a version of the efficient

markets hypothesis (EMH). A strict form of the efficient markets hypothesis states that information

observable to the market prior to period 𝑡 should not help to predict the return during period 𝑡. If

the (EMH) is false, then returns might be predictable. If so, then returns will be autocorrelated, and

the autoregressive model will provide useful forecasts.

For example, if you want to find out if returns are predictable (even if it is just a bit), estimate the

following AR(1)

𝑅𝑡 = 𝛽0 + 𝛽1 𝑅𝑡−1 + 𝑢𝑡+1

•

•

•

A positive 𝛽1 coefficient means “momentum,” past “good returns” mean higher future returns.

A negative 𝛽1 coefficient means “overreaction” or “mean reversion”. In this case, previous

“good returns” mean lower future returns.

Either way, if 𝛽1 ≠ 0, then returns will be autocorrelated, and the autoregressive model will

provide useful forecasts.

Note: In all your calculations use Huber-White heteroskedasticity consistent standard errors and

covariance.

a. Repeat the calculations reported in Table 15.2, using the following regression specifications

estimated over the 1960:M1–2002:M12 sample period.

AR(1) Model

𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝑒𝑡

AR(2) Model

𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝛽2 𝑟𝑡−2 + 𝑒𝑡

AR(4) Model

𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝛽2 𝑟𝑡−2 + 𝛽3 𝑟𝑡−3 + 𝛽4 𝑟𝑡−4 + 𝑒𝑡

Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–2002:M12

Dependent variable: Excess returns on the CRSP value-weighted index

Specification

Regressors

(1)

AR(1)

(2)

AR(2)

(3)

AR(4)

Excess Ret(t-1)

Std. Error

p-value

Excess Ret(t-2)

Std. Error

p-value

Excess Ret(t-3)

Std. Error

p-value

Excess Ret(t-4)

Std. Error

p-value

Intercept

Std. Error

p-value

Adj R^2

Wald F-statistic

p-value

T=

b. Are these results consistent with the theory of efficient capital markets?

c. Can you provide an intuition behind this result?

d. Repeat the calculations reported in Table 15.6, using regressions estimated over the 1960:M1–

2002:M12 sample period.

Autoregressive Distributed Lag Models of Monthly Excess Stock Returns, 1960:M1–2002:M12

Dependent variable: Excess returns on the CRSP value-weighted index

(1)

(2)

Specification

ADL(1,1)

ADL(2,2)

Eatimation Period

1960:M1–2002:M12

1960:M1–2002:M12

Regressors

Excess Ret(t-1)

Std. Error

p-value

(3)

ADL(1,1)

1960:M1–1992:M12

Excess Ret(t-2)

Std. Error

p-value

Change_ln_DP(t-1)

Std. Error

p-value

Change_ln_DP(t-2)

Std. Error

p-value

ln_DP(t-1)

Std. Error

p-value

Intercept

Std. Error

p-value

Adj R^2

F-statistic

p-value

Obs =

e. Does the Δln(dividend yield) have any predictive power for stock returns?

f.

Does “the level of the dividend yield” have any predictive power for stock returns?

g. Construct pseudo out-of-sample forecasts of excess returns over the 1993:M1–2002:M12 period,

using the regression specifications below that begin in 1960:M1.

ADL(1,1) specification:

𝐸𝑅𝑡 = 𝛽0 + 𝛽1 𝐸𝑅𝑡−1 + 𝛽2 𝑙𝑛(𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑)𝑡−1 + 𝑢𝑡

Constant Forecast: (in which the recursively estimated forecasting model includes only an

intercept)

𝐸𝑅𝑡 = 𝛽0 + 𝑢𝑡

Zero Forecast: the sample RMSFEs of always forecasting excess returns to be zero.

Model

Zero Forecast

RMSFE

Constant Forecast

ADL(1, 1)

h. Does the ADL(1,1) model with the log dividend yield provide better forecasts than the zero or

constant models?

Part 2

Forecasting models for the rate of inflation – Guidelines

Go to FRED’s website (https://fred.stlouisfed.org/) and download the data for:

•

Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) – Seasonally adjusted –

Monthly Frequency – From 1947:M1 to 2017:M12

In this hands-on exercise you will construct forecasting models for the rate of inflation, based on

CPIAUCSL.

For this analysis, use the sample period 1970:M01–2012:M12 (where data before 1970 should be used, as

necessary, as initial values for lags in regressions).

a.

(i)

Compute the (annualized) inflation rate,

(ii)

Plot the value of Infl from 1970:M01 through 2012:M12. Based on the plot, do you think

that Infl has a stochastic trend? Explain.

(i)

Compute the first twelve autocorrelations of (𝐼𝑛𝑓𝑙 𝑎𝑛𝑑 Δ𝐼𝑛𝑓𝑙)

(ii)

Plot the value of Δ𝐼𝑛𝑓𝑙 from 1970:M01 through 2012:M12. The plot should look

“choppy” or “jagged.” Explain why this behavior is consistent with the first

autocorrelation that you computed in part (i) for Δ𝐼𝑛𝑓𝑙.

(i)

Compute Run an OLS regression of 𝐼𝑛𝑓𝑙𝑡 on 𝐼𝑛𝑓𝑙𝑡−1. Does knowing the inflation this

month help predict the inflation next month? Explain.

(ii)

Estimate an AR(2) model for Infl. Is the AR(2) model better than an AR(1) model?

Explain.

(iii)

Estimate an AR(p) model for 𝑝 = 0, … ,8. What lag length is chosen by BIC? What lag

length is chosen by AIC?

(iv)

Use the AR(2) model to predict “the level of the inflation rate” in 2013:M01—that is, 𝐼𝑛𝑓𝑙2013:𝑀01 .

(i)

Use the ADF test for the regression in Equation (14.31) with two lags of 𝛥𝐼𝑛𝑓𝑙 to test for

a stochastic trend in 𝐼𝑛𝑓𝑙.

b.

c.

d.

(ii)

Is the ADF test based on Equation (14.31) preferred to the test based on Equation (14.32)

for testing for stochastic trend in 𝐼𝑛𝑓𝑙? Explain.

(iii)

In (i) you used two lags of 𝛥𝐼𝑛𝑓𝑙. Should you use more lags? Fewer lags? Explain.

(iv)

Based on the test you carried out in (i), does the AR model for 𝐼𝑛𝑓 contain a unit root?

Explain carefully. (Hint: Does the failure to reject a null hypothesis mean that the null

hypothesis is true?)

e. Use the QLR test with 15% trimming to test the stability of the coefficients in the AR(2) model

for “the inflation” 𝐼𝑛𝑓𝑙. Is the AR(2) model stable? Explain.

f.

(i)

Using the AR(2) model for 𝐼𝑛𝑓𝑙 with a sample period that begins in 1970:M01, compute

pseudo out-of-sample forecasts for the inflation beginning in 2005:M12 and going

through 2012:M12.

(ii)

Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a

nonzero mean?

(iii)

How large is the RMSFE of the pseudo out-of-sample forecasts? Is this consistent with

the AR(2) model for 𝐼𝑛𝑓𝑙 estimated over the 1970:M01–2005:M12 sample period?

(iv)

There is a large outlier in 2008:Q4. Why did inflation fall so much in 2008:Q4? (Hint:

Collect some data on oil prices. What happened to oil prices during 2008?)

Instructions:

• Please complete the guided project by May 8, 11:59 PM (ET).

• Write your answer below each question and upload a “word doc” named

LastName_FirstName.doc using the link on Blackboard.

• Also, you must upload the work files from R (LastName_FirstName.prg). One for

Part 1 and one for Part 2. Excel is not suitable for this project and it will not be

accepted.

This project consist of two parts:

• Part 1: Predicting Stock Returns.

• Part 2: Forecasting models for the rate of inflation.

Part 1: Predicting Stock Returns.

Data Description:

Documentation for Stock_Returns_1931_2002

This file contains 2 monthly data series over the 1931:1-2002:12 sample period.

ExReturn: Excess Returns

ln_DivYield (growth rate): 100×ln(dividend yield). (Multiplication by 100 means the

changes are interpreted as percentage points).

The data were supplied by Professor Motohiro Yogo of the University of Pennsylvania and

were used in his paper with John Campbell:

“Efficient Tests of Stock Return Predictability,” Journal of Financial

Economics, 2006. (Double click in the window below to access the data)

DATE_M

1931M01

1931M02

1931M03

1931M04

1931M05

1931M06

1931M07

1931M08

1931M09

EXRETURN

LN_DIVYIELD

5.9649584

-282.2329

10.3053054

-293.2089

-6.8408314

-287.8614

-10.4480653

-278.2477

-14.3580770

-265.4742

12.8502610

-280.5102

-6.6559179

-275.5950

0.0461485

-278.4424

-34.2583894

-247.1829

Some Background

exreturn: is the excess return on a broad- based index of stock prices, called

the CRSP value -weighted index, using monthly data from 1960:M1 to

2002:M12, where “M1” denotes the first month of the year (January), “M2”

denotes the second month, and so forth.

•

The monthly excess return is what you earn, in percentage terms,

by purchasing a stock at the end of the previous month and selling

it at the end of this month, minus what you would have earned had

you purchased a safe asset (a U.S. Treasury bill). The return on the

stock includes the capital gain (or loss) from the change in price

plus any dividends you receive during the month.

Calculating k-period stock returns:

One-period holding return:

Two-period holding return:

Other way

Three-period’s returns:

k-period’s returns:

When to apply a “buy and hold” strategy:

•

•

•

•

•

If you have a reliable “forecast” of future stock returns then an active “buy and hold”

strategy will make you rich quickly by beating the stock market.

If you think that the stock market will be going up, you should buy stocks today and sell

them later, before the market turns down. Forecasts based on past values of stock returns

are sometimes called “momentum” forecasts: If the value of a stock rose this month,

perhaps it has momentum and will also rise next month.

If so, then returns will be autocorrelated, and the autoregressive model will provide

useful forecasts. You can implement a momentum-based strategy for a specific stock or

for a stock index that measures the overall value of the market.

From another point of view, we can use autoregressive models to test a version of the

efficient markets hypothesis (EMH). A strict form of the efficient markets hypothesis

states that information observable to the market prior to period should not help to

predict the return during period . If the (EMH) is false, then returns might be

predictable. If so, then returns will be autocorrelated, and the autoregressive model will

provide useful forecasts.

For example, if you want to find out if returns are predictable (even if it is just a bit),

estimate the following AR(1)

•

•

•

A positive

coefficient means “momentum,” past “good returns” mean higher

future returns.

A negative

coefficient means “overreaction” or “mean reversion”. In this case,

previous “good returns” mean lower future returns.

Either way, if

, then returns will be autocorrelated, and the autoregressive

model will provide useful forecasts.

Note: In all your calculations use Huber-White heteroskedasticity consistent standard errors and

covariance.

a. Repeat the calculations reported in Table 14.2, using the following regression

specifications estimated over the 1960:M1–2002:M12 sample period.

AR(1) Model

AR(2) Model

AR(4) Model

Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–

2002:M12

Dependent variable: Excess returns on the CRSP value-weighted

index

(1)

(2)

(3)

Specification

AR(1)

AR(2)

AR(4)

Regressors

Excess Ret(t-1)

0.05

0.053

0.054

Std. Error

0.051

0.051

0.051

p-value

0.327

0.297

0.297

Excess Ret(t-2)

Std. Error

p-value

-0.053

0.048

0.274

-0.054

0.048

0.265

Excess Ret(t-3)

Std. Error

p-value

0.009

0.05

0.853

Excess Ret(t-4)

Std. Error

p-value

-0.016

0.047

0.736

Intercept

Std. Error

p-value

0.312

0.198

0.116

0.328

0.199

0.1004

0.33

0.203

0.105

Adj R^2

0.001

0.001

-0.002

Wald F-statistic

p-value

0.965

0.327

1.334

0.264

0.7

0.592

T=

516

516

516

b. Are these results consistent with the theory of efficient capital markets?

Yes, these results are consistent with the theory of efficient capital market because the P-value in

Excess Ret(t-1) is greater than 0.05. This shows that the AR (1) model is hardly useful in

forecasting. It is also greater than 0.05 in AR (2) also indicating that the respective models are

not useful for forecasting. However, it is less than 0.05 significant level for both Excess Ret(t-3)

at 0.009 and Excess Ret(t-4) at 0.016. These results are also consistent with the theory because it

holds that excess returns can be unpredictable.

c. Can you provide an intuition behind this result?

An intuition behind the results is that the autoregressive model of the monthly excess stock

returns is hardly a good predictor of stock price returns. The results are in line with efficient

financial markets hypothesis which holds that there are no statistically significant coefficients in

the estimated models. In this regard, we cannot reject the hypotheses that the coefficients are

equal to zero. Since the Adjustment R2 is almost to zero in the three models with a negative for

AR(4) model, we can resonate that the models are not good predictors of stock returns.

d. Repeat the calculations reported in Table 14.6, using regressions estimated over the

1960:M1– 2002:M12 sample period.

Autoregressive Distributed Lag Models of Monthly Excess Stock Returns, 1960:M1–

2002:M12

Dependent variable: Excess returns on the CRSP value-weighted index

(1)

(2)

Specification

ADL(1,1)

ADL(2,2)

Eatimation Period

1960:M1–2002:M12

1960:M1–2002:M12

Regressors

Excess Ret(t-1)

0.059

0.042

Std. Error

0.158

0.162

p-value

0.71

0.798

Excess Ret(t-2)

Std. Error

p-value

Change_ln_DP(t-1)

Std. Error

p-value

(3)

ADL(1,1)

1960:M1–1992:M12

0.078

0.057

0.17

-0.213

0.193

0.237

0.009

0.157

0.957

Change_ln_DP(t-2)

Std. Error

p-value

-0.012

0.163

0.944

-0.161

0.185

0.386

ln_DP(t-1)

Std. Error

p-value

0.026

0.012

0.025

Intercept

Std. Error

p-value

0.309

0.199

0.123

0.372

0.208

0.076

8.987

3.912

0.015

Adj R^2

-0.001

-0.001

0.013

F-statistic

p-value

Obs =

0.625

0.521

516

0.897

0.466

516

3.683

0.026

396

e. Does the

have any predictive power for stock returns?

The

lacks the predictive power for stock returns since the p-values are

higher than 0.05. This means that the null hypothesis is not rejected. Typically, the models of the

t-statistics tend to point towards the notion that the respective coefficients are not zero

f. Does “the level of the dividend yield” have any predictive power for stock returns?

Yes, “the level of the dividend yield” has predictive power for stock returns. The p-value

is 0, which necessitates us to reject the null hypothesis.

g. Construct pseudo out-of-sample forecasts of excess returns over the 1993:M1–

2002:M12 period, using the regression specifications below that begin in 1960:M1.

ADL(1,1) specification:

Constant Forecast: (in which the recursively estimated forecasting model includes only an

intercept)

Zero Forecast: the sample RMSFEs of always forecasting excess returns to be zero.

Model

RMSFE

Zero Forecast

3.99

Constant Forecast

3.98

ADL(1, 1)

4.30

h. Does the ADL(1,1) model with the log dividend yield provide better forecasts than

the zero or constant models?

I would say no. The ADL(1,1) model with the log dividend yield does not provide better

forecasts than the zero or constant models. With regards to the results, the model performs

poorer based on the RMSFE sample with respect to the intercept-only function. A model

forecasting that excess returns would be zero would further register a lower sample RMSFE.

Part 2

Forecasting models for the rate of inflation – Guidelines

Go to FRED’s website (https://fred.stlouisfed.org/) and download the data for:

Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) – Seasonally

adjusted – Monthly Frequency – From 1947:M1 to 2017:M12

In this hands-on exercise you will construct forecasting models for the rate of inflation,

based on CPIAUCSL.

For this analysis, use the sample period 1970:M01–2012:M12 (where data before 1970 should

be used, as necessary, as initial values for lags in regressions).

a.

(i)

Compute the (annualized) inflation rate.

(ii)

22.445

Plot the value of Infl from 1970:M01 through 2012:M12. Based on the plot, do

you think that Infl has a stochastic trend? Explain.

Based on the plot, it is probable that inflation has a stochastic trend since the

trend for the next day can hardly be predicted. This can only mean that the

trend is statistically random.

b.

(i)

Compute the first twelve autocorrelations of

From R-results

(ii)

Plot the value of

from 1970:M01 through 2012:M12. The plot should look

“choppy” or “jagged.” Explain why this behavior is consistent with the first

autocorrelation that you computed in part (i) for

.

The behavior is consistent with the autocorrelation that I computed in part (i) for

because

the longer bars towards the edges for some years depict higher volatility compared to the shorter

bars in the middle which depict stability.

c.

(i)

Compute Run an OLS regression of

on

. Does knowing the

inflation this month help predict the inflation next month? Explain.

Based on the results, it is imparative to agree that knowing the inflation in this

month helps predict the inflation in next month. This follows the fact that p =

0, and hence the null hypothesis is worth rejecting. This means that there is

predictive power for the next month.

(ii)

Estimate an AR(2) model for Infl. Is the AR(2) model better than an AR(1)

model? Explain.

Based on the results, the AR(2) model is much better than an AR(1) model because the Rsquare for AR(2) is greater than that of AR(1).

(iii) Estimate an AR(p) model for p =0, …, 8 . What lag length is chosen by BIC?

What lag length is chosen by AIC?

From R-results: Both AIC and BIC have lag lengths of 2 each.

d.

(i)

Use the AR(2) model to predict “the level of the inflation rate” in 2013:M01—

that is,

.

From R-results, the level of the inflation rate in 2013 is 0.017743 or 1.77%.

(ii)

Is the ADF test based on Equation (14.31)

preferred to the test based on Equation (14.32) for testing for stochastic trend in

? Explain.

From R-results, we can conclude that it is preferred since when the p-value is increases, it will

not be significant once the lag period is add and it is not significant.

(iii)

In (i) you used two lags of

. Should you use more lags? Fewer lags?

Explain.

More lags should not be used because it is not possible to add more variables owing to the fact

that the value is not that much significant.

(iv)

Based on the test you carried out in (i), does the AR model for

contain a unit

root? Explain carefully. (Hint: Does the failure to reject a null hypothesis mean

that the null hypothesis is true?)

Based on the test in (i), the AR model for

does not contain a unit root since the t-statistic in

this case is -2.3 and the p = 0.1724. This means that the null hypothesis cannot be rejected.

e. Use the QLR test with 15% trimming to test the stability of the coefficients in the

AR(2) model for “the inflation” Infl . Is the AR(2) model stable? Explain.

Based on the R-results, we can ascertain that the AR(2) model is not stable because the QLR is

29.7 with a p-critical value of about 6.02. This implies that we should reject the null hypothesis

since at least one of the coefficients changes over time.

f.

(i) Using the AR(2) model for

with a sample period that begins in 1970:M01, compute

pseudo out-of-sample forecasts for the inflation beginning in 2005:M12 and going

through 2012:M12.

(ii) Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a

nonzero mean?

I would say yes that the pseudo out-of-sample forecasts are biased. The biasness value is 0.279.

(iii)

How large is the RMSFE of the pseudo out-of-sample forecasts? Is this consistent

with the AR(2) model for

estimated over the 1970:M01–2005:M12 sample

period?

Accordingly, RMSFE is 0.020259 and is consistent with the AR(2) model for

over the 1970:M01–2005:M12 sample period.

(iv)

estimated

There is a large outlier in 2008:Q4. Why did inflation fall so much in

2008:Q4? (Hint: Collect some data on oil prices. What happened to oil prices

during 2008?)

From R-results, the rate of inflation fell significantly 2008. Inflation fell so much because of

the 2008 financial crisis and its impact of the global economy, which lead to the witnessed

economic decline as evident through a serious drop in oil prices.

TABLE 15.2 Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–2002:M12

Table 15.6 Autoregressive Distributed Lag Models of Monthly Excess Stock Returns

## We've got everything to become your favourite writing service

### Money back guarantee

Your money is safe. Even if we fail to satisfy your expectations, you can always request a refund and get your money back.

### Confidentiality

We don’t share your private information with anyone. What happens on our website stays on our website.

### Our service is legit

We provide you with a sample paper on the topic you need, and this kind of academic assistance is perfectly legitimate.

### Get a plagiarism-free paper

We check every paper with our plagiarism-detection software, so you get a unique paper written for your particular purposes.

### We can help with urgent tasks

Need a paper tomorrow? We can write it even while you’re sleeping. Place an order now and get your paper in 8 hours.

### Pay a fair price

Our prices depend on urgency. If you want a cheap essay, place your order in advance. Our prices start from $11 per page.