Testing the consumption-based CAPM using the stochastic discount factor

MONTEIRO, MARCEL STANLEI; CARRASCO-GUTIERREZ, CARLOS ENRIQUE

doi:10.5935/0034-7140.20220004

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Artigos • Rev. Bras. Econ. 76 (1) • Jan-Mar 2022 • https://doi.org/10.5935/0034-7140.20220004 linkcopiar

Testing the consumption-based CAPM using the stochastic discount factor^*

Autoria SCIMAGO INSTITUTIONS RANKINGS

This article investigates the problem of optimal intertemporal consumption in the CCAPM setup from a new empirical perspective. The econometric analysis is based on use of the equality between the stochastic discount factor (SDF) and the marginal rate of intertemporal substitution of consumption, which in the CCAPM is equivalent to the Euler equation resulting from the intertemporal optimization problem of the representative individual. We start from an asset pricing equation to find the estimators of the SDF, without the need to make a parametric assumption about preferences, and then estimate the parameters of the consumption models. In our empirical exercise, the dataset covers income, aggregate consumption and return on financial assets in the quarterly period from 1996:1 to 2016:4. We also consider the existence of a portion of rule-of-thumb consumers and the utility functions CRRA and habit formation in consumer preferences. The empirical results suggest that the preferences that exhibit the formation of consumption habits combined with the stochastic discount factor originating from the hypotheses of Brownian motion are those that most closely correspond to the hypotheses related to the behavior of aggregate consumption.

Model	Utility function	Stochastic discount factor
CRRA	$u (C_{2, t}) = \frac{({(C_{2, t})}^{1 - γ} - 1)}{(1 - γ)}$	$m_{t + 1} = [β {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$
Habit Formation	$u (C_{2 t}, C_{2, t - 1}) = \frac{1}{1 - γ} {(\frac{C_{2, t}}{C_{2, t - 1}^{k}})}^{1 - γ}$	$m_{t + 1} = [β {(\frac{C_{t} - λ Y_{t}}{C_{t - 1} - λ Y_{t - 1}})}^{k (γ - 1)} {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$
CRRA		In logarithmic terms ln m_t+1 = ln β − γΔ [ln(C_t+1 − λY_t+1)]
Habit Formation		ln m_t+1 = ln β − k(γ − 1) Δ [ln(C_t+1 − λY_t)] − γΔ [ln(C_t+1 − λY_t+1)]

	SDF-BM	SDF-HJ	SDF-CAPM	Correlation	SDF-BM	SDF-HJ	SDF-CAPM
Mean	0.78	0.94	0.98	SDF-BM	1.000
Median	0.75	0.91	0.98	SDF-HJ	-0.377	1.000
Maximum	1.36	1.54	1.32	SDF-CAPM	0.839	-0.574	1.000
Minimum	0.22	0.62	0.61
Std. Dev.	0.23	0.19	0.14

	CRRA: $u (C_{2, t}) = \frac{{(c_{2, t})}^{1 - γ} - 1}{1 - γ}$						External Habit: $u (C_{2 t}, C_{2, t - 1}) = \frac{1}{1 - γ} {(\frac{C_{2, t}}{C_{2, t - 1}^{k}})}^{1 - γ}$
	$m_{t + 1} = [β {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − γΔ [ln(C_t+1 − λY_t+1)]			$m_{t + 1} = [β {(\frac{C_{t} - λ Y_{t}}{C_{t - 1} - λ Y_{t - 1}})}^{k (γ - 1)} {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − k(γ − 1) Δ [ln(C_t+1 − λY_t)] − γΔ [ln(C_t+1 − λY_t+1)]
	(I)			(II)			(I)			(II)
	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6
β	0.7931*** (0.0258)	0.7851*** (0.0255)	0.7810*** (0.0255)	0.7586*** (0.0272)	0.7506*** (0.0263)	0.7467*** (0.0260)	0.8032*** (0.0267)	0.7954*** (0.0253)	0.7809*** (0.02570)	0.7694*** (0.0288)	0.7641*** (0.0267)	0.7466*** (0.0261)
γ	3.1869* (1.7895)	1.6370* (0.9275)	-0.0218 (0.1060)	2.9965 (1.9378)	1.372 (1.1132)	-0.028 (0.1123)	3.3722* (1.7914)	2.3789** (0.9032)	-0.0096 (0.1081)	3.0711 (1.9314)	2.4658** (1.1759)	-0.0138 (0.1149)
k		-	-	-	-	-	-0.9352 (1.0063)	-1.3013 (0.9311)	0.0519 (0.1071)	-1.1701 (1.4090)	-1.5301 (1.2117)	0.0712 (0.1153)
ARCH Test¹	8.5496*	5.5882	8.4329*	1.8771	1.7209	2.1605	7.691585	4.446435	7.4668	1.9296	1.7548	2.0991
LM Test²	2.3064	3.5597	3.703	0.8984*	1.8165	1.4511	3.123008	4.735853	3.496	1.202	2.638	1.4032
Jarque-Bera³	1.2215	1.3451	2.1766	16.477***	16.123***	16.653***	1.207257	1.526151	2.2992	17.209***	11.170***	15.192***

	CRRA: $u (C_{2, t}) = \frac{{(c_{2, t})}^{1 - γ} - 1}{1 - γ}$						External Habit: $u (C_{2 t}, C_{2, t - 1}) = \frac{1}{1 - γ} {(\frac{C_{2, t}}{C_{2, t - 1}^{k}})}^{1 - γ}$
	$m_{t + 1} = [β {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − γΔ [ln(C_t+1 − λY_t+1)]			$m_{t + 1} = [β {(\frac{C_{t} - λ Y_{t}}{C_{t - 1} - λ Y_{t - 1}})}^{k (γ - 1)} {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − k(γ − 1) Δ [ln(C_t+1 − λY_t)] − γΔ [ln(C_t+1 − λY_t+1)]
	(I)			(II)			(I)			(II)
	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6
β	0.9368*** (0.2195)	0.9424*** (0.0213)	0.9405*** (0.0273)	0.9202*** (0.0206)	0.9244*** (0.0202)	0.9237*** (0.0197)	0.9338*** (0.0231)	0.9418*** (0.0220)	0.9399*** (0.0206)	0.9171*** (0.0216)	0.9228*** (0.0209)	0.9236*** (0.0197)
γ	-0.8871 (1.2567)	0.4752 (0.6975)	0.1090 (0.0686)	-0.7591 (1.2133)	0.3238 (0.6940)	0.0884 (0.0687)	-0.9120 (1.2620)	0.4265 (0.7626)	0.1292* (0.0687)	-0.7765 (1.2200)	0.2087 (0.7609)	0.0996 (0.0702)
k	-	-	-	-	-	-	-0.2731 (0.6763)	-0.2447 (1.3002)	0.1053 (0.0848)	-0.3187 (0.7093)	-0.3713 (0.8886)	0.0630 (0.0792)
ARCH Test¹	3.1252	2.7246	4.2501	3.1773	1.9352	3.0199	2.8937	3.0035	2.5674	3.1802	2.4359	2.2058
LM Test²	3.7501	4.0376	2.9404	3.9872	4.3067	3.3979	3.8774	3.8813	3.3203	4.1887	4.0752	3.5833
Jarque-Bera³	10.048***	8.6162**	6.2302**	1.0365	1.2140	0.8823	10.210**	8.4676**	3.9070	1.1911	1.4771	0.4304

	CRRA: $u (C_{2, t}) = \frac{{(c_{2, t})}^{1 - γ} - 1}{1 - γ}$						External Habit: $u (C_{2 t}, C_{2, t - 1}) = \frac{1}{1 - γ} {(\frac{C_{2, t}}{C_{2, t - 1}^{k}})}^{1 - γ}$
	$m_{t + 1} = [β {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − γΔ [ln(C_t+1 − λY_t+1)]			$m_{t + 1} = [β {(\frac{C_{t} - λ Y_{t}}{C_{t - 1} - λ Y_{t - 1}})}^{k (γ - 1)} {(\frac{C_{t + 1} - λ Y_{t + 1}}{C_{t} - λ Y_{t}})}^{- γ}]$			ln m_t+1 = ln β − k(γ − 1) Δ [ln(C_t+1 − λY_t)] − γΔ [ln(C_t+1 − λY_t+1)]
	(I)			(II)			(I)			(II)
	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6	λ = 0	λ = 0.3	λ = 0.6
β	0.9834*** (0.0167)	0.9805*** (0.0164)	0.9788*** (0.0161)	0.9728*** (0.0172)	0.9697*** (0.0168)	0.9679*** (0.0165)	0.9848*** (0.0176)	0.9850*** (0.0166)	0.9788*** (0.0162)	0.9731*** (0.0182)	0.9746*** (0.0172)	0.9679*** (0.0166)
γ	0.8952 (0.9255)	0.2806 (0.5226)	-0.2895 (0.0536)	0.9498 (0.9594)	0.3337 (0.5499)	-0.0256 (0.0550)	0.9075 (0.9322)	0.5062 (0.5518)	-0.0273 (0.0550)	0.9518 (0.9660)	0.6012 (0.5934)	-0.0237 (0.0565)
k	-	-	-	-	-	-	2.5839 (28.2087)	0.5834 (1.4885)	0.0093 (0.0529)	1.3379 (33.8693)	0.7629 (2.1348)	0.009 (0.0553)
ARCH Test¹	3.6638	3.2959	4.4861	8.3804*	7.9194*	9.3291*	3.7254	2.9212	4.3436	8.3589*	7.2487	9.2253*
LM Test²	3.8568	4.6171	3.3007	4.0793	4.9417	3.3409	4.0308	3.697	3.1803	4.1253	3.9259	3.232
Jarque-Bera³	0.0364	0.0628	0.2428	5.1114	5.1779*	6.5792**	0.0798	0.1178	0.2497	5.2661*	5.8812*	6.5713**

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[TFN1] Note: The parameter γ is the relative risk aversion coefficient and G is the parameter that governs the temporal separability of the utility function.

Testing the consumption-based CAPM using the stochastic discount factor*

Testing the consumption-based CAPM using the stochastic discount factor^*