Show That for the Following Regression Model $$Y _ { t } = e ^ { \beta _ { 0 } + \beta _ { 1 } x d + u }$$

Essay

Show that for the following regression model $$Y _ { t } = e ^ { \beta _ { 0 } + \beta _ { 1 } x d + u }$$
where $t$ is a time trend, which takes on the values $1,2 , \ldots , T , \beta _ { 1 }$ represents the instantaneous ("continuous compounding") growth rate. Show how this rate is related to the proportionate rate of growth, which is calculated from the relationship
$$Y _ { t } = Y _ { 0 } \times ( 1 + g ) ^ { t }$$
when time is measured in discrete intervals.