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research can be improved by incorporating sentimental analysis of text data from social media and other investment related websites.
Second, the six confidence indexes from J.P. Morgan Asset Management are quarterly data. To work with the weekly data from Google Trends and from the Taiwan Economic Journal, we have transformed the indexes into weekly data, which might not reflect investor’s sentiments precisely. As shown in this research, some of the indexes are not significantly related to the TAIEX average returns. This result can be improved when higher frequency data become available.
Last, it is not clear to us why SVI has predictive power for the TAIEX average returns of the first, third, fourth and fifth week but not the second week. There might be other factors of SVIs we need to consider when using it to represent investors’
attention.
4. Recommendations for Future Research
We outline three areas of research for future works:
o The incorporation of sentimental analysis of text data from social media, such as Facebook, Twitter, and other investment related websites to represent investors’ sentiments. Not only are these data easier to obtain through web crawler, they also provide more precise and timely information about
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investors’ sentiments. Consequently, more accurate results might be obtained with less time delay.
o The investigation of investors’ Google search behaviors in other financial markets. While we have identified investors in the Taiwan Stock Market mostly use company names, not ticker symbols, to conduct Google search for stocks related information, it is not clear if this behavior is unique to the Taiwan Stock Market. This is an interesting research question to be explored in the future.
o The exploration of investors’ buying and selling intentions revealed from their Google search behaviours. In particular, we will investigate expanded Google search keywords, in addition to company names or ticker symbols, and will use other tools such as Google Correlate to study investors behaviours in the Taiwan Stock Market (see Stocking & Matsa, 2017).
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Appendix
1. Newey-West Correction of Standard Errors Method Explained
1. Below is the proof of kernel HAC from Kuan (2008).
“Consider the linear specification 𝑦𝑦𝑖𝑖= 𝑥𝑥𝑖𝑖′𝛽𝛽 + 𝑒𝑒𝑖𝑖, and the OLS estimator 𝛽𝛽̂𝑇𝑇(𝑘𝑘 × 1). We shall
review some basic asymptotic theory for 𝛽𝛽̂𝑇𝑇 and the Wald test of regression parameters. In what follows,
we let [𝑐𝑐] denote the integer part of c, → convergence in probability, ⇒ week convergence (of ℙ
associated probability measures), → convergence in distribution 𝐷𝐷 =dequality in distribution, 𝑊𝑊𝑘𝑘 a vector
of 𝑘𝑘 independent, standard Wiener processes and 𝐵𝐵𝑘𝑘 the Brownian bridge obtained from 𝑊𝑊𝑘𝑘 such that
𝐵𝐵𝑘𝑘(𝑟𝑟) = 𝑊𝑊𝑘𝑘(𝑟𝑟) − 𝑟𝑟𝑊𝑊𝑘𝑘(1), 0≤ r ≤ 1. When 𝑘𝑘 = 1, we simply write 𝑊𝑊1 as 𝑊𝑊 and 𝐵𝐵1 as 𝐵𝐵.
We impose the following “high level” conditions on data.
[A1] For some 𝛽𝛽0, ε𝑖𝑖= 𝑦𝑦𝑖𝑖− 𝑥𝑥𝑖𝑖′𝛽𝛽0 such that 𝔼𝔼(𝑥𝑥𝑖𝑖𝜀𝜀𝑖𝑖) = 0 and
1
√𝑇𝑇∑|𝑇𝑇𝑖𝑖=1𝑟𝑟|𝑥𝑥𝑖𝑖𝜀𝜀𝑖𝑖⇒ 𝑆𝑆0𝑊𝑊𝑘𝑘(𝑟𝑟), r ∈ [0,1], where 𝑆𝑆0 is the nonsingular, matrix square root of
Σ0= lim
𝑇𝑇→∞𝑣𝑣𝑣𝑣𝑟𝑟(√𝑇𝑇1 ∑𝑇𝑇𝑖𝑖=1𝑥𝑥𝑖𝑖𝜀𝜀𝑖𝑖) , i.e., Σ0= 𝑆𝑆0𝑆𝑆0′.
[A2] M𝑇𝑇 = 𝑇𝑇−1∑|𝑇𝑇𝑖𝑖=1𝑟𝑟|𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖′→ 𝑀𝑀ℙ 0 uniformly in r ∈ (0,1] such that 𝑀𝑀0 is nonsingular.
By [A1],
‧
This is the well-known asymptotic normality result for the OLS estimator.
With the results in (1), the limiting distributions of the well-known large-sample tests are easily
obtained. Consider the null hypothesis 𝐻𝐻0: 𝑅𝑅𝛽𝛽0= 𝑟𝑟 with R a q × 𝑘𝑘 matrix with full row rank. Under
the null hypothesis, (1) implies
√𝑇𝑇�𝑅𝑅𝛽𝛽̂𝑇𝑇− 𝑟𝑟�→ 𝒩𝒩(0, 𝑅𝑅M𝐷𝐷 0−1Σ0M0−1𝑅𝑅′). (2)
Replacing 𝑀𝑀0 and Σ0 with their respective consistent estimators 𝑀𝑀𝑇𝑇 and Σ�𝑇𝑇, we have
(𝑅𝑅MT−1Σ�𝑇𝑇MT−1𝑅𝑅)−1/2√𝑇𝑇𝑅𝑅�𝛽𝛽̂𝑇𝑇− 𝛽𝛽0�→ 𝒩𝒩�0, 𝐼𝐼𝐷𝐷 𝑞𝑞�.
It follows that the Wald test of this hypothesis is
𝑊𝑊𝑇𝑇 = 𝑇𝑇�𝑅𝑅𝛽𝛽̂𝑇𝑇− 𝑟𝑟�′(𝑅𝑅MT−1Σ�𝑇𝑇MT−1𝑅𝑅′)−1�𝑅𝑅𝛽𝛽̂𝑇𝑇− 𝑟𝑟�→ 𝒳𝒳𝐷𝐷 2(𝑞𝑞). (3)
Note that 𝑊𝑊𝑇𝑇 would not have a limiting 𝒳𝒳2 distribution if Σ�𝑇𝑇 is not a consistent estimator for Σ0.
For other large-sample tests, such as the LM test and Hausman test, it is also crucial to have a consistent
estimator of the asymptotic variance-covariance matrix.
2. HAC Estimators
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For consistent estimation of Σ0, first note that, by definition,
Σ0= lim
For notation simplicity, we write
Σ0= lim
3. Kernel HAC Estimators
It is clear that exact form of Σ0 depends on data characteristics. When 𝑥𝑥𝑖𝑖𝜀𝜀𝑖𝑖 are serially
uncorrelated, all the autocovariances in (4) vanish, so that
Σ0= lim
𝑇𝑇→∞Γ𝑇𝑇(0) = lim𝑇𝑇→∞1𝑇𝑇∑𝑇𝑇𝑖𝑖=1𝔼𝔼(𝜀𝜀𝑖𝑖2𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖′). (5) The variance-covariance matrix can be consistently estimated by White’s heteroscedasticity consistent
estimator:
Σ�𝑇𝑇=1𝑇𝑇�𝑇𝑇𝑖𝑖=1𝑒𝑒̂𝑖𝑖2𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖′,
‧
denote a function of that diverges with T we have
Σ𝑇𝑇+= � 𝑙𝑙(𝑇𝑇) Γ𝑇𝑇(𝑗𝑗) →
proper variance-covariance matrix. A consistent estimator that is also positive semi-definite is the
following estimator of the spectral density:
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Σ�𝑇𝑇𝑘𝑘 = Σ𝑗𝑗=−𝑇𝑇+1𝑇𝑇−1 κ �𝑙𝑙(𝑇𝑇)𝑗𝑗 � Γ�𝑇𝑇(𝑗𝑗), (6)
where κ is a proper kernel function and 𝑙𝑙(𝑇𝑇) is its bandwidth, which jointly determine the weights
assigned to Γ�𝑇𝑇(𝑗𝑗). Typically, κ is required to satisfy: |𝜅𝜅(𝑥𝑥)| ≤ 1, 𝜅𝜅(0) = 1, 𝜅𝜅(𝑥𝑥) = 𝜅𝜅(−𝑥𝑥) for all 𝑥𝑥 ∈
ℝ, ∫|𝜅𝜅(𝑥𝑥)| 𝑑𝑑𝑥𝑥 < ∞, κ is continuous at 0 and at all but a finite number of other points in ℝ, and
� 𝜅𝜅(𝑥𝑥)∞
−∞ 𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑥𝑥 ≥ 0, ∀ω ∈ ℝ.
Note that the last condition ensures positive semi-definiteness; see Andrews (1991).
Below is Parzen kernel (Gallant, 1987):
1 − 6𝑥𝑥2+ 6|𝑥𝑥|3, |𝑥𝑥| ≤12
κ(𝑥𝑥) = 2(1 − |𝑥𝑥|)3, 12≤ |𝑥𝑥| ≤ 1
0, otherwise
”
2. Code
Data paper;input date log_search_volume_change log_AvgPrice_change log_volume_change
mood_price_change mood_Tweconomic_change mood_government_change mood_possibility_change mood_global_change mood_investment_change;
datalines;
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… …
;
proc model data=paper;
exog log_search_volume_change log_volume_change mood_price_change mood_Tweconomic_change ;
instruments _exog_;
parms b0 b1 b2 b3 b4 ;
log_AvgPrice_change = b0 + b1*log_search_volume_change + b2*log_volume_change + b3*mood_price_change +
b4*mood_TWeconomic_change;
fit log_AvgPrice_change / gmm kernel=(parzen,4,0) vardef=n;
run;
quit;
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