On the Predictability of Stock Prices: A Case for High and Low Prices

Caporin, Massimiliano; Ranaldo, Angelo; Magistris, Paolo Santucci de

doi:10.2139/ssrn.1866223

“…Under the constraint d = d 0 , the profile log-likelihood function ℓ T (ψ) only varies with respect to b and it has a unique maximum around b 0 . Interestingly, Lemma 1 provides theoretical support to the procedure, adopted in Bollerslev et al (2013) and Caporin et al (2013), of estimating the FCVAR d,b model by restricting the fractional parameter d to a constant value and by maximizing the profile log-likelihood function with respect to b only. Figure 6 reports the value of the sliced profile log-likelihood with respect to different values of b, when the parameter d is fixed at the true value d 0 = 1.…”

Section: Lemma 1 Letθmentioning

confidence: 82%

On the Identification of Fractionally Cointegrated VAR Models With theF(d)Condition

Carlini

¹

,

Magistris

²

2017

Journal of Business & Economic Statistics

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“…Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, that is, that d 0 −b 0 < 1∕2, and extending the results to allow d 0 −b 0 > 1∕2 requires new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Such non-stationary cointegrating vectors have been found in many empirical studies; some examples in finance using the fractional CVAR model are Caporin et al (2013, Table 2), Barunik and Dvorakova (2015, Table 6), and Dolatabadi et al (2016, Tables 5 and 6).…”

Section: Introductionmentioning

confidence: 81%

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Johansen

¹

,

Nielsen

²

2018

Journal Time Series Analysis

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We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non-stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non-fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi-squared-test. NONSTATIONARY COINTEGRATION IN THE FCVAR MODEL 521 2. ASSUMPTIONS AND MAIN RESULTSIn Johansen and Nielsen (2012a), asymptotic properties of maximum likelihood estimators and test statistics were derived for model (1) with the parameter space ≤ b ≤ d ≤ d 1 for some d 1 > 0, which can be arbitrarily large, and some such that 0 < ≤ 1∕2. The parameter space was extended by Johansen and Nielsen (2018) toagain for an arbitrarily large d 1 > 0 and an arbitrarily small such that 0 < ≤ 1∕2. While is exactly the same as in Johansen and Nielsen (2012a), we have in (3) introduced the new constant 1 > 0, which is zero in Johansen and Nielsen (2012a). We note that the parameter space  explicitly includes the linein the interior precisely because 1 > 0. Although > 0 can be arbitrarily small, a smaller implies a stronger moment condition for consistency in both Johansen and Nielsen (2012a) and Johansen and Nielsen (2018). This moment condition is relaxed below. We will assume that the data for t ≥ 1 are generated by model (1 ). A standard approach for autoregressive models, which we follow, is to conduct inference using the conditional likelihood function of X 1 , … , X T given initial values {X −n } n≥0 . That is, we interpret (1) as a model for X t , t = 1, … , T, given the past, and use the conditional density to build a conditional likelihood function. Thus, since our entire approach is conditional on the initial values {X −n } n≥0 , we consider these non-random, as is standard for (especially non-stationary) autoregressive models.However, it is difficult to imagine a situation where {X s } T s=−∞ are available, or perhaps even exist, so we assume that the data are only observed for t = −N +1, … , T. Johansen and Nielsen (2016) argue in favor of the assumption that series was initialized in the finite past using two leading examples, political opinion poll data and financial volatility data, but we maintain the more general assumption from J...

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“…JN(2012a) assumed that the cointegrating vectors are stationary, i.e., that d 0 −b 0 < 1/2, and extending the results to allow d 0 −b 0 > 1/2 requires new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Such nonstationary cointegrating vectors have been found in many empirical studies; some examples in finance using the fractional CVAR model are Caporin et al (2013, Tables 5-6). …”

Section: Introductionmentioning

confidence: 86%

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Johansen

¹

,

Nielsen

²

2018

View full text Add to dashboard Cite

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are nonstationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non-fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a χ 2 -test.

show abstract

On the Predictability of Stock Prices: A Case for High and Low Prices

Cited by 16 publications

References 45 publications

On the Identification of Fractionally Cointegrated VAR Models With theF(d)Condition

On the Identification of Fractionally Cointegrated VAR Models With theF(d)Condition

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

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