Question

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 11.27 (p. 657). Recall that opening weekend box office revenue data (in millions of dollars) were collected for a sample of 24 recent movies. In addition to each movie’s tweet rate, i.e., the average number of tweets referring to the movie per hour 1 week prior to the movie’s release, the researchers also computed the ratio of positive to negative tweets (called the PN-ratio).

a) Give the equation of a first-order model relating revenue $\left(y\right)$to both tweet rate$\left(x_1\right)$and PN-ratio$\left(x_2\right)$.

b) Which b in the model, part a, represents the change in revenue$\mathbf{\left(}\mathbf{y}\mathbf{\right)}$for every 1-tweet increase in the tweet rate$\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}$, holding PN-ratio$\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$constant?

c) Which b in the model, part a, represents the change in revenue $\mathbf{\left(}\mathbf{y}\mathbf{\right)}$for every 1-unit increase in the PN-ratio$\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$, holding tweet rate$\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}$constant?

d) The following coefficients were reported:${\mathit{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{945}$and${{\mathit{R}}_{\mathbf{a}}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{940}$. Give a practical interpretation for both${\mathit{R}}^{\mathbf{2}}$and${{\mathit{R}}_{\mathbf{a}}}^{\mathbf{2}}$.

e) Conduct a test of the null hypothesis, ${\mathit{H}}_{\mathbf{0}}\mathbf{;}{\mathit{\beta }}_{\mathbf{1}}\mathbf{=}{\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$. Use$\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.

f) The researchers reported the p-values for testing,${\mathit{H}}_{\mathbf{0}}\mathbf{;}{\mathit{\beta }}_{\mathbf{1}}\mathbf{=}\mathbf{0}$and${\mathit{H}}_{\mathbf{0}}\mathbf{;}{\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$ as both less than .0001. Interpret these results (use).

Short answer

a) The first order model equation is$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon$

b) In part a, ${\beta }_1$represents change in revenue $\left(y\right)$for 1-unit increase in the tweet rate $\left(x_1\right)$while ${\beta }_2$represents change in revenue $\left(y\right)$for 1-unit increase in PN-ratio $\left(x_2\right)$.

c) ${\beta }_2$represents change in revenue $\left(y\right)$for 1-unit increase in PN-ratio $\left(x_2\right)$while holding tweet rate $\left(x_1\right)$constant.

d) Both $R^2$and ${R_a}^2$values are near 94%, the model is a good fit for the data.

e) We reject the null hypothesis

f) p-values for testing, $H_0:{\beta }_1=0$and $H_0:{\beta }_2=0$is less than .0001 then for $\alpha =0.01$, we reject the $H_0$since p-value <$\alpha$ .

Step-by-step solution

First order model equation

The first order model equation for revenue to both tweet rate and PN-ratio is

$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon$

Where tweet rate is denoted by $x_1$and $x_2$is denoted by PN-ratio

Slopes of  β1and β2

In part a, ${\beta }_1$represents change in revenue $\left(y\right)$for 1-unit increase in the tweet rate $\left(x_1\right)$while ${\beta }_2$represents change in revenue role="math" localid="1651880604187" $\left(y\right)$for 1-unit increase in PN-ratio$\left(x_2\right)$.

Slope of independent variable while other independent variables are held constant

In part a, ${\beta }_2$represents change in revenue $\left(y\right)$for 1-unit increase in PN-ratio $\left(x_2\right)$while holding tweet rate $\left(x_1\right)$constant.

 R2and adjusted R2

Values of $R^2$and ${R_a}^2$is 0.945 and 0.940 indicating that almost 90% of the variation in the variables can be explained by the model. The model is a good fit for the data.$R^2$ just indicates a value that tell us if the model is a good fit for the data or not while adjusted $R^2$adjusts for the no of independent variables present in the model and explains if they are good fit for the data or not.

Since both these values are near 94%, the model is a good fit for the data.

Overall significance of the model 

We reject the null hypothesis

Interpretation of p-values

p-values for testing, $H_0:{\beta }_1=0$and $H_0:{\beta }_2=0$is less than .0001 then for $\alpha =0.01$, we reject the $H_0$since p-value <$\alpha$ .