代写ECON5163 Financial Econometrics: Problem Set 2025-2026帮做Python编程

ECON5163

Financial Econometrics: Problem Set

2025-2026

In this individual problem set, the aim is to study the theory and empirical testing of the Con-sumption Capital Asset Pricing Model (CCAPM). The tasks include deriving the model, developing the linear and nonlinear empirical specifications, and estimating them by Ordinary Least Squares (OLS) and the Generalized Method of Moments (GMM) using real data. The results should then be interpreted in economic terms to assess whether there is evidence of the equity premium puzzle and to discuss possible solutions. There are five questions to be addressed, with grading weights indicated in parentheses.

Please refer to the assignment guidelines on page 5 for further details.

1. Theory and Derivation: The CCAPM links asset returns to consumption growth, un-like the traditional CAPM which links them to the market portfolio. Intuitively, when the consumption path (or consumption growth) is very volatile agents want to invest to smooth consumption. This higher demand for assets makes them more expensive (prices increase). Key differences between the models are:

• CCAPM derives from utility maximization and intertemporal choice, while CAPM is based on mean–variance optimization.

• CCAPM emphasizes consumption risk, whereas CAPM emphasizes market risk

• CCAPM allows for a multi-period framework, CAPM is essentially a single-period model.

The first task is to derive the Euler equation and the linear approximation for empirical estimation. Assume the existence of a representative investor who chooses to consume and invest in a single asset (a stock index) so that at each time t he maximises his expected life-time utility:

subject to the intertemporal budget constraint:

At = Rt(At−1 + Yt−1 − Ct−1), ∀t ≥ 0

where:

• At is is the amount of the financial asset held by the representative consumer

• Yt is the labour income

• Rt is the real gross rate of return on the asset

• Ct is the real consumption

• 0 < β ≤ 1 is the discount factor

• E0 is the expectation operator at time 0

In turn, Rt is defined as follows:

where Pt is the ex-dividend real stock price index and Dt is the real aggregate dividend received between time t − 1 and t.

(a) Consider a two period setting with t = 0, 1. Assume no consumption, saving, or labor income after t = 1. Set A0 = 0 and Y1 = 0 and assume all remaining wealth is consumed at t = 1, so A1 = C1. Preferences are constant relative risk aversion (CRRA) with per period utility:

where γ ≠ 1 is the coefficient of relative risk aversion.1 Rewrite the maximization prob-lem and the intertemporal budget constraint for this two period case, then substitute C1 from the constraint into the objective function. Differentiate the objective with respect to C0 using the chain rule and set the first order condition equal to zero, simplify the expression, and where useful substitute for C1 again. Provide a step by step derivation that leads to the Euler equation below:

(b) The CCAPM Euler equation can be written more generally as:

Under rational expectations this condition needs to hold in conditional expectation, not exactly each period. Define the period t deviation εt and write:

Take logs of both sides, then apply the first order approximation ln(1 + εt) ≈ εt for small εt to obtain a log linear form. Show all steps to obtain:

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2. Dataset: The goal is to construct a dataset suitable for estimating the CCAPM model, requiring data on Pt, Dt, and Ct. Use the following sources:

• Real stock prices and dividends are available at monthly frequency from Professor Robert Shiller’s website (Yale University).

• Real per capita consumption is available at quarterly frequency from FRED (Federal Re-serve Bank of St. Louis): “Real personal consumption expenditures per capita: Goods: Nondurable goods (A796RX0Q048SBEA)”.

To construct the dataset, proceed as follows:

• Convert Shiller’s monthly data to quarterly frequency by retaining the last observation of each quarter.

• Compute the gross real return as

and define the log real return as rt = ln(Rt).

• Define log real consumption as ct = ln(Ct) and its growth as ∆ct = ct − ct−1.

• Keep only dates that appear in all series and truncate the sample at 2020Q1, removing all observations after 2020Q1.

Report summary statistics for the key variables in Table 1. Plot Rt, ct, and ∆ct, and discuss whether these series fluctuate around a constant mean.

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3. Linear Estimation: The objective is to estimate the log-linear approximation of the CCAPM model derived in Question 1, expressed as:

rt = θ + γ∆ct + εt

where θ ≡ − ln(β).

(a) Estimate this equation by OLS using robust standard errors. From the estimated inter-cept ˆθ, recover ˆβ. Report ˆβ and ˆγ along with their p-values in Table 2.

(b) State the assumptions required for consistency of the OLS estimator, and write the formulas for the estimators expressed with the variables in this model.

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4. Non-linear Estimation: The objective is to estimate the non-linear CCAPM model derived in Question 1.

(a) Estimate this Euler equation by GMM using instruments available at time t−1. Include a constant and lags of consumption growth and gross returns as instruments. Consider the sets Zt−1 = {1, ∆ct−1, Rt−1}, and Zt−2 = {1, ∆ct−1, ∆ct−2, Rt−1, Rt−2}. Implement two-step GMM with a heteroskedasticity-consistent (HC) covariance matrix, and perform. the Hansen J test of overidentifying restrictions. From the estimated coefficients, recover ˆβ and ˆγ with p-values.4 Record the results in Table 2. For numerical stability, implement the model using the rewritten equivalent form. of the Euler equation:

where β = e−θ.

(b) Write the sample moment condition for GMM using the instrument set Zt−1, expressed with the variables in this model. Then explain how two-step GMM estimation works, and discuss the purpose of the Hansen J test of overidentifying restrictions, including how to interpret its p-value.

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5. Interpretation and Analysis:

(a) Compare the estimates reported in Table 2 and identify which specification provides the most appropriate results, explaining your choice. Discuss whether the estimates for ˆβ and ˆγ lie within a plausible range, and justify your answer.

(b) Using the estimated coefficient of relative risk aversion ˆγ, evaluate whether the results provide evidence of the equity premium puzzle. Briefly explain what the puzzle refers to and discuss which feature of the sample data might account for the estimated value of risk aversion.

(c) Present one approach from the literature that addresses the equity premium puzzle, and provide the reference. Briefly explain the modification to the standard model and why it enables the observed equity premium to be reconciled with more plausible values of γ.

Table 2: Estimates of the CCAPM for U.S. data. Numbers in parentheses are p-values. For OLS, robust standard errors are used. For GMM, two-step estimation with a heteroskedasticity-consistent (HC) covariance matrix is applied. The J-test is the Hansen test of overidentifying restrictions. Parameters: β denotes the discount factor and γ the coefficient of relative risk aversion. The instrument sets are: Zt−1 = {1, ∆ct−1, Rt−1} and Zt−2 = {1, ∆ct−1, ∆ct−2, Rt−1, Rt−2}.

Guidelines:.

• Include your Python or R script. in the Appendix (the appendix does not count toward the page limit).

• Submit a single document containing all written answers, tables, figures, and the code ap-pendix.

• The page limit is 5 pages (excluding tables, figures, and appendix). Any content exceeding this limit by more than 10% will not be considered.

• Suggested references:

– Azar, S. 2011, “Retesting the CCAPM Euler equations”, International journal of man-agerial finance, vol. 7, no. 4, pp. 324.

– Breeden, D.T., Gibbons, M.R. & Litzenberger, R.H. 1989, “Empirical Test of the Con-sumption-Oriented CAPM”, The Journal of finance (New York), vol. 44, no. 2, pp. 231-262.

– Lund, J. & Engsted, T. 1996, “GMM and present value tests of the C-CAPM: evidence from the Danish, German, Swedish and UK stock markets”, Journal of international money and finance, vol. 15, no. 4, pp. 497-521.

– Mehra, R. & Prescott, E.C. 1985, “The equity premium: A puzzle”, Journal of monetary economics, vol. 15, no. 2, pp. 145-161.