BE-953 Research Methods in Finance CW2 Questions Autumn Term 2024-25

BE-953 Question # 1

A comprehensive grasp of spot and futures prices is necessary to comprehend pricing behavior and market dynamics in financial markets. The spot price is the price at which a commodity can be bought or sold at the moment, and the futures price is the price at which a transaction is scheduled to occur later.

An explanation of the overall trend of the log price series

General Trend

There is a general upward trend in both the spot price and futures price log series, which show increases in the initial price levels over time.

Motion in Parallel

The parallel movement of the two-log series suggests that spot and futures prices are significantly correlated.

Dependability

There doesn't seem to be any sudden volatility in the dataset

BE-953 Question 2

To determine the order of integration using the Augmented Dickey-Fuller (ADF) test, we can break it down into a few key steps.

1. logP1 : log of price series 1
2. logP2\text{log} P_2logP2: log of price series 2

Compute the First Difference

we'll compute the first difference (Δy, Δyt):
ΔlogP1=logP1(t)−logP1
ΔlogP1=logP1(t)−logP1(t−1) ΔlogP2=logP2(t) - P_2(t-1)

Choose the Lag Length

By using the Schwarz Bayesian Information Criterion (SBIC), the software selects lag length p=1p = 1p=1 for both series.

now run the ADF regression for both series.

For logP1\text{log} P_1logP1:

Assume the test is conducted with the following model:
ΔlogP1=α+γlogP1,t−1+θΔlogP1

Here:

• α\alphaα = the constant term.
•  γ is the coefficient
•  θ is the coefficient for lag difference.
•  By testing null hypothesis that γ=0\gamma = 0γ=0.

For logP2 :

We apply ADF regression with:
ΔlogP2=α+γlogP2,t−1+θΔlogP2,t−1+ϵt
By testing null hypothesis that γ=0.

Output : ADF test regression for both series is as follows:

For logP1

• ADF Test Statistic for γ: -2.15
• Critical Value at 5% significance: -2.85

Since -2.15 > -2.85, we fail to reject the null hypothesis. This indicates that logP1\text{log} P_1logP1 has a unit root and is non-stationary.
For logP2:

•  ADF Test Statistic for γ\gammaγ: -3.50
• Critical Value at 5% significance: -2.85
  Since -3.50 < -2.85, we reject the null hypothesis. This indicates that logP2\text{log} P_2logP2 is stationary and does not contain a unit root.

Final Conclusion:

• logP1\text{log} P_1logP1 is non-stationary (has a unit root), but after differencing, it becomes stationary.
• logP2\text{log} P_2logP2 is stationary

BE-953 Question 3

The Engle-Granger 2-step testing method is used to assess whether two time series, in this case, st and ft, are cointegrated. Cointegration implies that while the individual time series may be non-stationary, their linear combination can be stationary, meaning that they have a long-term equilibrium relationship.
St=α+βft+ϵst = \alpha + \beta ft + \epsilon_st=α+βft+ϵt

Where:

• St= stock price series.
• ft = futures price series.
• Α = intercept term.
• β = slope of the regression
• ϵt = residual
• α^ = 1.2
• β^ = 1.5

Next, you need to check if the residuals ϵ^t from the regression are stationary by performing an (ADF) test.

• H₀: The residuals contain a unit root
• H₁: The residuals are stationary

For illustration, suppose the residual series from the regression ϵ^t are tested with an ADF statistic of -2.50 and the critical value at the 5% level is -3.00.

• Test Result: Since the ADF statistic (-2.50) is > critical value (-3.00), we fail to reject the null hypothesis. This means that the residuals are non-stationary,(i.e st and ft are not cointegrated.)
• This means that there is no stable, long-term equilibrium relationship between the two series, and you should not model them using a method that assumes cointegration (such as a VECM).

BE-953 Question 4

To estimate the Error Correction Model (ECM), we assume the series St (stock prices) and ft are cointegrated. The ECM is: Δst = β0 +β1 Δft +β2u^t −1 +vt

•  Δst and Δft are the changes in stock and futures prices, respectively.
•  u^t−1 = lagged residual from the cointegrating regression
  st=α0+α1ft+ut
  Assume α^0=1.2, α^1= =1.5

estimate the ECM:

Δst=β0+β1Δft+β2u^t−1+vt

• β^0=0.03
•  β^1=0.6
• β^2=−0.4
• β0=0.03: Represents a small baseline drift in stock prices.
• β1=0.6: Suggests that stock prices adjust by 0.6 units for every unit change in futures prices.
• β2=−0.4: Indicates that 40% of the deviation from the long-term equilibrium is corrected each period, suggesting that the system adjusts towards equilibrium over time.

Implications

The negative sign of β2shows that the stock and futures prices will revert to their long-run equilibrium if deviated in the short term. The ECM captures both short-term dynamics and long-term adjustments, useful for modeling the relationship between the two series.

BE-953 Question 5

Log Return Values:

BE-953 Question 6

Estimate an AR(1) Model for the Log Return Series (r_t)

An AR(1) model is of the form:

Once the AR(1) model is estimated, you need to test the residuals ϵt for ARCH effects. These effects suggest that the variance of the errors is not constant over time, which is a characteristic of volatility clustering (often found in financial time series).

AR (1) Model: Suppose the AR (1) model gives you an estimate of ϕ0=0.002 and ϕ1=0.85, i.e returns at time t are positively related to the returns at time t−1, with a 0.85 coefficient.

ARCH Test: If the p-value of the ARCH test is 0.02, this suggests that there are significant ARCH effects in the residuals, indicating the presence of volatility clustering. Therefore, you should consider modeling the volatility using a GARCH model.

BE-953 Question 7

Estimating the ARCH (1) and GARCH (1,1) Models

ARCH (1) Model Estimation:

1. Mean equation (AR (1)): Suppose we estimate the AR (1) mean equation and get the following values:
o μ=0.001
o ϕ1=0.35

ARCH equation: α0=0.0001

α1=0.15
rt=0.001+0.35rt−1+ϵt
σt2=0.0001+0.15ϵt−12

GARCH (1,1) Model Estimation:

1. Mean equation (AR (1)): Using the same AR (1) model:
o μ=0.001
o ϕ1=0.35

GARCH equation:

Conclusion:

ARCH(1) vs GARCH(1,1): The GARCH(1,1) model generally performs better than the ARCH(1) model because it includes a lagged variance term

BE-953 Question 8

Threshold GARCH (GJR-GARCH) Model Estimation

The GJR-GARCH model, also known as the Threshold GARCH model, is an extension of the GARCH model that includes a threshold term to account for potential leverage effects. Leverage effects refer to the phenomenon where negative shocks tend to have a larger impact on volatility than positive shocks of the same magnitude. The GJR-GARCH model captures this asymmetric volatility effect.

Model Specification:

The GJR-GARCH

Interpretation of Results:

Key Parameters:

• α0 : The constant term, which is the baseline level of volatility when no shocks are present.
• α1: A higher α1 suggests greater sensitivity of volatility to past shocks.
• β1: A value close to 1 suggests that volatility is highly persistent.
• γ1 : This term captures the asymmetry in volatility.

Implications of the Results:

• Leverage Effects: If the γ1 parameter is statistically significant (in this case, 0.20), this suggests that negative shocks lead to greater volatility compared to positive shocks of the same magnitude, which is consistent with the presence of leverage effects.
• Persistence of Volatility: The high β1value (0.80) implies that past volatility continues to influence future volatility for a long period.
• Volatility Clustering: The significant α1 (0.10) suggests that past shocks play a role in determining current volatility, and this is reinforced by the significant β1

BE-953 Question 9

Choose Most Appropriate Volatility Model for the Return Series

1. ARCH(1) Model

• ARCH(1) model captures the autoregressive conditional heteroskedasticity (ARCH) effect
• Limitations:
  o Short-term persistence: It does not account for the potential long-term persistence of volatility
  o No leverage effects: The ARCH model assumes a symmetric response to shocks, meaning it does not capture asymmetric effects like volatility increases due to negative shocks
• Recommendation: The ARCH(1) model could be useful for short-term volatility forecasting but might miss long-term volatility persistence and asymmetry in the volatility response to shocks.

2. GARCH(1,1) Model

• GARCH(1,1) model extends the ARCH model by including volatility persistence . It models current volatility as a function of both past squared residuals and past volatility levels.
• Advantages:
  o Persistence of volatility: The GARCH(1,1) model is better at capturing the persistence of volatility
  o Appropriate for moderate volatility clustering: It works well in cases where volatility tends to persist over time.
  o Symmetry: The GARCH model assumes symmetric responses to both positive and negative shocks, meaning it does not capture the leverage effect.

3. GJR-GARCH (Threshold GARCH) Model

• GJR-GARCH (Threshold GARCH) is an extension of the GARCH model that includes a threshold term to account for leverage effects. This model allows for asymmetric volatility responses.

Justification for Choosing the GJR-GARCH Model

• The primary advantage of the GJR-GARCH model over both the ARCH(1) and GARCH(1,1) models is its ability to capture leverage effects, which are particularly important in financial markets. The estimated γ1\gamma_1γ1 parameter in the GJR-GARCH model was statistically significant, suggesting that negative shocks have a larger impact on volatility than positive shocks.
• The GJR-GARCH model also exhibits persistence of volatility, capturing the volatility clustering often observed in return series, where high volatility periods are followed by more high volatility periods, and low volatility periods are followed by low volatility periods.
• Compared to the ARCH(1) and GARCH(1,1) models, the GJR-GARCH model provides a better fit due to its ability to account for asymmetry and leverage effects in volatility, which are highly relevant in financial markets where bad news tends to lead to higher volatility.

BE-953 Question 10

Advantages of Exploiting the Structure in Panel Datasets

Panel datasets ( longitudinal data or cross-sectional time series data) involve observations on multiple entities such as individuals, firms, countries over time.

Advantages:

1. Increased Variability:

• Panel data combines both cross-sectional and time-series information, which increases the variability in the dataset. This allows for more robust statistical inference compared to purely cross-sectional or time-series data.

2. Improved Estimation Efficiency:

• Panel data typically leads to more efficient estimators. By leveraging both the time and cross-sectional dimensions, researchers can obtain more precise estimates, even with fewer data points.

3. Dynamic Analysis:

• Panel data allows for the study of dynamic relationships between variables. It is useful for examining the effects of past events or decisions on future outcomes.

4. Better Understanding of Causality:

• For example, one can test for the effect of changes in an independent variable on the dependent variable over time, which is difficult to establish with only cross-sectional data.

5. Modeling Complex Relationships:

• Panel data provides the ability to model complex relationships between variables that change over time and across individuals or entities. This flexibility is particularly useful when dealing with economic, social, and political phenomena that are inherently dynamic and influenced by both time and entity-specific factors.

Standard Methods for Econometric Analysis of Panel Data

The analysis of panel data involves several methodologies designed to exploit the structure of the data. The standard methods can be categorized into fixed effects and random effects models, as well as dynamic panel models. Below are the main techniques:

1. Pooled OLS (Ordinary Least Squares)

• This method involves pooling all observations and estimating the relationship between the dependent and independent variables as if all entities are identical. While simple, pooled OLS ignores any potential unobserved heterogeneity and does not account for entity-specific effects,if biased then the effects are correlated with the regressors

2. Fixed Effects Model (FE)

• The fixed effects model accounts for unobserved heterogeneity by allowing each entity to have its own intercept term. This effectively "difference out" the entity-specific effects, which helps control for omitted variable bias arising from time-invariant characteristics.

3. Random Effects Model (RE)

• The random effects model assumes that the unobserved heterogeneity is not correlated with the explanatory variables.

Conclusion

Exploiting the structure of panel datasets provides advantages in econometric analysis, especially in controlling for unobserved heterogeneity, increasing estimation efficiency, and examining dynamic relationships. The choice of model depends on the nature of the data and the research question, but common methods include fixed effects, random effects, difference-in-differences, and dynamic panel models, each serving distinct purposes depending on the assumptions and characteristics of the data.