### 统计代写|风险建模代写Financial risk modeling代考|The statistical significance of the economic gains

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## 统计代写|风险建模代写Financial risk modeling代考|The statistical significance of the economic gains

One way to assess the statistical significance of the economic gains resulting from Tables $1.7-1.9$ is to perform the following joint statistical test. For any target $\mu_{p}$ and any estimator, one can define alternative covariance forecasts $\hat{C}{t}$ and portfolio returns $R{t+1}^{p(C)}$. Define
$$a_{t+1}^{\hat{C}}=\left(R_{t+1}^{p(\text { Fourier })}-\bar{R}^{p(\text { Fourier })}\right)^{2}-\left(R_{t+1}^{p(\mathcal{C})}-\bar{R}^{p(\hat{C})}\right)^{2} .$$
Assessing the statistical significance of the economic gains of the Fourier estimate over alternative forecasts can be conducted by testing whether the mean of $a_{t+1}^{\hat{C}}$ is larger than (or equal to) zero against the alternative that the mean is smaller than zero. Following Bandi et al.

(2006), for any target return $d=0.09,0.12,0.15$, we define the vector
$$A_{t+1}^{d}=\left(a_{t+1}^{\hat{C}{1}}, a{t+1}^{\hat{C}{2}}, \ldots, a{t+1}^{\hat{C}{r}}\right)^{\prime},$$ where the $r$-uple of estimators $\left(\hat{C}{1}, \hat{C}{2}, \ldots, \hat{C}{r}\right)$ is given by $\left(R C^{1 \min }, R C^{5 \mathrm{~min}}\right.$, $\left.R C^{10 m i n}\right),\left(R C L L^{1 m i n}, R C L L^{5 m i n}, R C L L^{10 m i n}\right)$ and (RCopt $\left., A O, K, A O_{s u b}\right)$ or any other combination of methods we want to test. We also stack all the methods simultaneously and check the overall ability of the Fourier method to yield a significant economic gain over the others. We write the regression model
$$A_{t+1}^{d}=\delta^{d} \mathbf{1}{r}+\varepsilon{t+1},$$
where $\delta^{d}$ is a scalar parameter. Series $a_{t+1}^{\hat{C}}$ associated to losses (i.e. negative values in Tables $1.7-1.9$ ) are multiplied by $-1$ before regression. We perform the one-sided test $H_{0}: \delta^{d} \geq 0$, against $H_{A}: \delta^{d}<0$. The parameter $\delta^{d}$ is estimated by GMM using a Bartlett HAC covariance matrix. A similar approach is used by Engle and Colacito (2006). The $t$-statistics of all the tests imply rejection of the null hypothesis, and hence statistical significance of the economic gains/losses at the 5 percent level. In particular, we remark that when testing the different methods altogether $(r=10)$ we get rejection of the null hypothesis even if we do not change the sign of the series $a_{t+1}^{\hat{C}}$ associated to losses. Indeed, in this case the corresponding $t$-statistics are $-5.69,-4.44$ and $-7.30$, respectively, revealing that on average the Fourier methodology yields a statistically significant economic gain at the 1 percent level.

## 统计代写|风险建模代写Financial risk modeling代考|Conclusion

We have analyzed the gains offered by the Fourier estimator from the perspective of an asset-allocation decision problem. The comparison is extended to realized covariance-type estimators, to lead-lag bias corrections, to the all-overlapping estimator, to its subsampled version and to the realized kernel estimator.

We show that the Fourier estimator carefully extracts information from noisy high-frequency asset-price data and allows for nonnegligible utility gains in portfolio management. Specifically, our simulations show that the gains yielded by the Fourier methodology are statistically significant and can be economically large, while only the subsampled alloverlapping estimator and, for low levels of market microstructure noise, the realized covariance with one lead-lag bias correction and suitable sampling frequency can be competitive. Analyzing the in-sample and out-of-sample properties of different covariance measures, we find that for increasing values of microstructure noise the Fourier estimator continues to provide precise variance/ covariance estimates which translate into more precise forecasts with respect to the other estimators under consideration, $A O_{s u b}$ being the only competitive method.

## 统计代写|风险建模代写Financial risk modeling代考|References

Aitt-Sahalia, Y. and Mancini, L. (2008) “Out of Sample Forecasts of Quadratic Variations, “Joumal of Econometrics, 147 (1): 17-33.
Ait-t-Sahalia, Y., Mykland, P. and Zhang. L. (2005) “How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise,” Review of Financial Studies, 18 (2): 351-416.
Andersen, T. and Bollerslev, T. (1998) “Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts, ” International Economic Review, 39 (4): 885-905.
Bandi, F.M. and Russell, J.R. (2006) “Separating Market Microstructure Noise from Volatility, ” Joumal of Financial Economics, 79 (3): $655-692$.
Bandi, F.M. and Russell, J.R. (2008) “Microstructure Noise, Realized Variance and Optimal Sampling,” Review of Economic Studies, 75 (2): 339-369.
Bandi, F.M., Russel, J.R. and Zhu, Y. (2008) “Using High-frequency Data in Dynamic Portfolio Choice,” Econometric Reviews, 27 (1-3): 163-198.
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (2008a) “Multivariate Realised Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices with Noise and Non-synchronous Trading, ” Economics Series Working Paper No 397, University of Oxford, Oxford, United Kingdom.
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (2008b) “Designing Realized Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise,” Econometrica, 76 (6): 1481-1536.
Barucci, E., Magno, D. and Mancino, M.E. (2008) “Forecasting Volatility with High Frequency Data in the Presence of Microstructure Noise,” Working paper, University of Firenze, Firenze, Italy.
De Pooter, M., Martens, M. and van Dijk, D. (2008) “Predicting the Daily Covariance Matrix for S\&P100 Stocks Using Intraday Data: But Which Frequency to Use?” Econometric Reviews, 27 (1): 199-229.
Engle, R. and Colacito, R. (2006) “Testing and Valuing Dynamic Correlations for Asset Allocation,” Journal of Business \& Economic Statistics, 24 (2): 238-253.

## 统计代写|风险建模代写Financial risk modeling代考|The statistical significance of the economic gains

(2006)，对于任何目标回报d=0.09,0.12,0.15，我们定义向量

## 统计代写|风险建模代写Financial risk modeling代考|References

Aitt-Sahalia, Y. 和 Mancini, L. (2008)“二次变分的样本预测之外”，“计量经济学杂志”，147 (1): 17-33。
Ait-t-Sahalia, Y.、Mykland, P. 和张。L. (2005) “在存在市场微观结构噪声的情况下多久对连续时间过程进行采样”，金融研究评论，18 (2): 351-416。
Andersen, T. 和 Bollerslev, T. (1998) “回答怀疑论者：是的，标准波动率模型确实提供了准确的预测，” 国际经济评论，39 (4): 885-905。
Bandi, FM 和 Russell, JR (2006) “从波动性中分离市场微观结构噪声”，金融经济学杂志，79 (3)：655−692.
Bandi, FM 和 Russell, JR (2008) “微观结构噪声、实现方差和最优抽样”，经济研究评论，75 (2): 339-369。
Bandi, FM, Russel, JR 和 Zhu, Y. (2008) “在动态投资组合选择中使用高频数据”，计量经济学评论，27 (1-3): 163-198。
Barndorff-Nielsen, OE, Hansen, PR, Lunde, A. 和 Shephard, N. (2008a) “Multivariate Realized Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of the Covariation with Noise and Non-synchronous Trading”，经济学系列第 397 号工作文件，牛津大学，英国牛津。
Barndorff-Nielsen, OE, Hansen, PR, Lunde, A. 和 Shephard, N. (2008b) “设计已实现的内核以测量存在噪声时股票价格的事后变化”，计量经济学，76 (6)： 1481-1536。
Barucci, E.、Magno, D. 和 Mancino, ME（2008 年）“Forecasting Volatility with High Frequency Data in the Presence of Microstructure Noise”，工作论文，意大利佛罗伦萨大学。
De Pooter, M.、Martens, M. 和 van Dijk, D.（2008 年）“使用盘中数据预测 S\&P100 股票的每日协方差矩阵：但使用哪个频率？” 计量经济学评论，27 (1): 199-229。
Engle, R. 和 Colacito, R. (2006) “Testing and Valuing Dynamic Correlations for Asset Allocation”，商业与经济统计杂志，24 (2): 238-253。

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