STA501 Mock exam – F2023

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\begin{center}
	\textbf{\Large STA502: Math \& Stat for MBA I \\ \bigskip Mock Exam F2023}\\
\end{center}

\begin{question}
	As a recent graduate of SIAI's renowned MBA in AI/BigData, you just got hired at one of the Fortune 500 companies. As with all rapidly growing tech companies, the compensation package is incredibly generous with handsome stock options, but it also has reputation of egocentric seniors in all levels. On the very first day, at a town-hall meeting, you were able to witness two founding members of the company, still young and energetic, in heated discussion in the perspective of benefits and costs of running an online community for AI specific contents. \\
	
	Boss A claims that the community needs more members to grow fast, and by the network effect, the number will grow, at some point, exponentially, which will eventually credit the company as an "AI expert's company". On the other hand, Boss B argues that the community needs more contents, especially supreme quality contents, to attract more people. Boss B thus insists that the company hire reputed data scientists across the states. However, since Boss A believes number of people pre-defines the community's recognition, he suggested to invite entry software engineers with little to no knowledge in scientific aspect of AI/Data Science. He continued that there are thousands of millions of people learning entry language called PyR, thus the community should provide basic libraries written in PyR to attract them. Boss B, conversely, claimed that advanced languages like Julian, despite limited user base, should be the company's focus to earn 'AI Expert's company' badge. \\
	
	After watching the impassioned debate, your data science team's boss gave you an assignment to verify which variable is a key to winning online community, between a large user base($U$) and amount of quality contents ($C$). \\
	
	Since you believe, for a successful community, both user and contents bases are simultaneous necessity and sufficiency, you wonder if this analysis has to incorporate simultaneity model that you learned in Math \& Stat for MBA. Still unsure what model can recover true nature of successful and reputed community, you have designed following two simple relationships: 
	\begin{align*}
		C &= \alpha_1 + \alpha_2 U + \alpha_3 M_2 + u \hspace{1cm} (1)\\
		U &= \beta_1 + \beta_2 C + v  \hspace{2.1cm} (2) 
	\end{align*}
	where $M_2$ may be assumed to be an exogenous variable for anything related to a successful community and $u$ and $v$ are identically and independently distributed disturbance terms with zero means. The observations for $M$ are drawn from a fixed population with finite mean and variance.
	\begin{enumerate}
		\item[1.] Derive the reduced form equation for $C$. (5 marks) \\
		
		\item[2.] Demonstrate that the OLS estimator of $\beta_2$ is, in general, inconsistent. How is your conclusion affected in the special case $\alpha_2 = 0$? How is your conclusion affected in the special case $\alpha_2 \beta_2 = 1$? What do these special case mean in words? (5 marks) \\
		
		\item[3.] Demonstrate that the instrumental variables (IV) estimator of $\beta_2$, using $M_2$ as an instrument for $C$, is consistent. Why do you need an IV estimator? (5 marks)\\
		
		\item[4.] Instead of using IV estimation, the researcher decides to use 2-Stage-Least-Square (2SLS) in the expectation of obtaining a more efficient estimator of $\beta_2$. He fits the reduced form equation for $C$: 
		\begin{align*}
			\hat{C} = k_1 + k_2 M_2 \hspace{2cm} (3)
		\end{align*}
		saves the fitted values, and uses them as an instrument for $U$ in equation (2). Demonstrate that the 2SLS estimator is consistent. (5 marks) \\
		
		\item[5.] Determine whether the researcher is correct in believing that the 2SLS estimator is more efficient than the IV estimator. (5 marks) \\
		
		\item[6.] How do you prove that IV (or 2SLS) estimation is superior to OLS? (5 marks)  \\
		
		\item[7.] If you have $M_1$ for equation (2), as is $M_2$ for equation (1), can you have any better result? If so, in what context? Can you argue more instruments promise better results? (5 marks) \\
	\end{enumerate}
	
	At that point, your data science team's boss asked your interim report, and you told him that you are looking for pertinent instrumental variable sets to strengthen your argument. You boss, however, is a strongly dis-believer of scientific models and has a firm belief on machines. He asks you to complete the research as soon as possible just by applying all machine learning models from PyR library and choose the most matching model. \\

	\begin{enumerate}		
		\item[8.] Can you extend your logic in 6) to disprove a claim that machine learning model is superior to OLS? Assume that your model's errors, even after 1st-stage data pre-processing, follow Gaussian distribution jointly. What happens if non-Gaussian? (5 marks) \\
		
		\item[9.] Having been benchpressed by your logic, your boss, with a firm belief on machine learning, claims that adding a quadratic term, instead of IV or 2SLS, is far more superior estimation strategy, because he believes non-linear \& non-parametric estimation by computers are better than human's faulty logical thinking, as was witnessed by Alpha-Go and abundant achievements by "Artificial Intelligence". Provide your rebuttal. (10 marks) \\ 
	\end{enumerate}
\end{question}


\begin{question}
	Since year 2040, when the government Sirius announced full scholarship to all domestic AI/Data Science programs in universities, there has been on-going debate whether the education actually is rewarding, in terms of quality and future wage of labor. Not all companies have vigorously hired AI grads, and after 15 years of record, one researcher wants to clarify if AI adoption by active hiring really helped companies to grow faster. \\
	
	The following regressions are for 9,125 AI/Data Science graduates in 2050. The data science strategy of the study is to compare various outcomes that helped companies to grow (assets, sales, and number of employees) for AI-trained and no-AI-trained in 2055. The regressions also interact AI grads' status with labor market's appreciation of the quality which is reflected in wage growth, $W_g$. Assume that the labor market of the country is efficient, that higher wage means higher productivity, at least in the field of data science. Wage growth is coded so that growth of 7\% would be 0.07. \\

	\begin{figure}[ht!]
		\centering
		\begin{tabular}{p{5cm}|ccc}
			& \multicolumn{3}{c}{Dependent variable}\\
			\hline
			& $\underset{(1)}{ln(Assets)}$ & $\underset{(2)}{ln(Sales)}$ & $\underset{(3)}{ln(Employment)}$\\
			\hline
			$D_h$ & $\underset{(0.027)}{0.089}$ & $\underset{(0.026)}{-0.131}$ & $\underset{(0.015)}{-0.108}$\\[5pt]
			$D_h \times W_g$ & $\underset{(0.18)}{1.21}$ & $\underset{(0.17)}{0.94}$ & $\underset{(0.11)}{0.37}$ \\
			$W_g$ & $\underset{(0.28)}{0.58}$ & $\underset{(0.26)}{0.29}$ & $\underset{(0.22)}{0.21}$ \\
			\hline
		\end{tabular}
	\end{figure}
	where $D_h$ is for Dummy for AI grads, and $W_g$ is for wage growth rate for AI grads. Standard errors are displayed in parentheses. All regressions also contain a constant term.	
	\vspace{0.2cm}	

	
	\begin{enumerate}
		\item[1.] Explain why a simple regression of business outcomes on the AI-training alone may not answer the question data scientists are interested in. (5 marks)\\
		
		\item[2.] Explain how the use of wage growth of AI grads may circumvent the problem you described in 1). What's the interaction term's function in words? (5 marks)\\ 
		
		\item[3.] Explain verbally what the coefficient of 0.089 on the dummy for AI-grads in column (1) means. (5 marks) \\
		
		\item[4.] If wage growth is 10 percentage points higher, how much higher are the sales of no-AI-trained companies in the sample on average? Explain whether this effect is statistically different from zero. (5 marks) \\
		
		\item[5.] What do you conclude from the results in the table about the effect of AI training on AI adopted company's outcomes? (5 marks) \\
		
		\item[6.] Suppose you also have data for assets, sales, and employment in these companies in 2060. Suppose you were to run analogous regressions with these dependent variables to the regressions in the table above. Explain how the new regressions would help you interpret the results above \\
		
		\item[7.] As more and more AI grads flow into the labor market, given the growing competition for top-minds, ranking services has been introduced to the market. The ranking service claims that they have differentiated AI education's quality for low tier ones like Engineering ($E$) and high tier ones like Science ($S$) in year 2055. Companies pay more to high tier grads, thus the wage growth rates are now $WE_g$ and $WS_g$ for Engineering and Science, respectively. How does this change affect your analysis in 6)? (5 marks) \\
		
		\item[8.] Given the change of regime in 2055, you would like to see whether the split of the program helped companies. How do you formulate your data scientific test?  (5 marks)\\
		
		\item[9.] You have an engineering background boss whose understanding of data science is no better than collection of Gitjjab codes. He claims that deep-learning can solve every data science problems that no human logic is needed. He adds that your argument does not rely on 'the most recent and advanced deep-learning practices done by top-notch companies and researchers'. Provide your rebuttal. (10 marks)\\
		
		\item[Bonus.] Assume that you are class of 2055-2056 at an engineering program. Back then, you were mis-guided by engineering school's marketing that they guarantee 100\% graduation rate and employment rate. In addition to that, you failed in admission exam to SIAI, one of the most well-known Science tier AI program in the world. Back then, you were so scared, but after years on the job, you realized that you have wasted your money and time. Now given 8), you have strong temptation to go back to school and re-try the Science tier. If successful, you can enjoy higher wage and better appreciation of the market. Given your personal estimation of success rate, formulate your argument. (10 marks)
	\end{enumerate}
\end{question}


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