Example 1: Gulati and Malhotra (2006) <\/a><\/h3><\/p>\nOn April 1, 1996, Canada and U.S. entered into a five-year contract called the Softwood Lumber Agreement (SLA<\/em> henceforth), whereby a tariff rate quota was imposed on US-bound lumber exports from four Canadian provinces: Alberta, British Columbia, Ontario and Quebec. The first 14.7 billion board feet of softwood lumber from these provinces was exported duty free but further exports were subject to tariffs. A tariff of $50 per thousand board feet was imposed on the next 650 million board feet exported and further exports were going to be taxed at $100 per thousand board feet. The SLA <\/em>was novel since exports from only four provinces were limited and other provinces were not subject to any restriction.<\/p>\nGulati and Malhotra<\/a> (2006) investigated firstly, whether the SLA<\/em> caused a reduction in softwood exports to the U.S. from the four provinces and if so, what was the size of the decline. Secondly, what was the size of the increases, if any, in the softwood lumber exports of the other provinces to the U.S.<\/p>\nThe regression equation they run is the following:<\/p>\n
Xit<\/sub> = \u03b10<\/sub> + \u03b11<\/sub>*Yit<\/sub> + \u03b12<\/sub>*YUS,t<\/sub> + \u03b13<\/sub>*Distt<\/sub> + \u03b14<\/sub>*Ext<\/sub> + \u03b15<\/sub>*RUS,t<\/sub> + \u03b16<\/sub>*SLA<\/em>i<\/sub> + \u03b17<\/sub>*Restt<\/sub> + \u03b18<\/sub>*SLA<\/em>i<\/sub>*Restt<\/sub> + ut<\/sub><\/p>\nand the STATA<\/abbr> command will be:<\/p>\nreg<\/strong> X Yi<\/sub> YUS<\/sub> Dist Ex R SLA<\/em> Rest SLA#<\/strong><\/em>Rest<\/p>\nwhere, Xit<\/sub> is log value of exports or log quantity of exports (annual) from province I to US. Yit<\/sub> and YUS,t<\/sub> are the log GDP of province i and US at time t respectively to control for demand of lumber in Canada and US. Disti<\/sub> is the log of distance from province I to the US border. RUS,t<\/sub> is the US interest rate to control for influence of interest rates on demand for new homes (a major source of softwood lumber). Ext<\/sub> is the US-Canada interest rate to control for price effects since exchange rates affect the relative price of Canadian lumber. SLA<\/em> is a dummy variable which takes the value 1 for provinces on which tariffs were applied under SLA<\/em> and 0 otherwise. Rest is a dummy for the years SLA<\/em> was in effect and 0 otherwise. SLA<\/em>*Rest is an interaction term that takes the value 1 for provinces under SLA<\/em> for the years SLA<\/em> was in place.<\/p>\nThe key coefficient of interest is \u03b18<\/sub> as it is interpreted as the causal effect of the SLA on the change in exports of the SLA<\/em> provinces compared to the non-SLA<\/em> provinces. To understand why \u03b18<\/sub> is referred as the \u201cDifference in Difference\u201d estimate, take a look at the following table which is Table 4 in the paper.<\/p>\n![\"\"](\"https:\/\/blogs.ubc.ca\/datawithstata\/files\/2019\/05\/did_table_gulati_malhotra-300x174.png\")
<\/p>\n
\u03b16<\/sub> captures the difference in the mean of log exports for SLA<\/em> and non-SLA<\/em> provinces before the SLA <\/em>restrictions came into effect. \u03b17<\/sub> captures the effect of the SLA <\/em>agreement on the non-SLA<\/em> provinces. \u03b17<\/sub> + \u03b18<\/sub> reflects the effect of the SLA<\/em> contract on the provinces named in the contract. Thus, \u03b18<\/sub> shows the difference in export performance of the provinces named in the SLA <\/em>vs. not included in the SLA<\/em> because of the SLA<\/em> restriction.<\/p>\nTheir results indicate that the SLA<\/em> had a significant impact on the exports of non-SLA<\/em> provinces and the SLA<\/em> by itself increased the exports from these provinces compared to the SLA<\/em> provinces by four times. But, the SLA<\/em> lead to a decrease in exports of SLA<\/em> provinces of only 5 percent which was statistically insignificant. Thus, SLA<\/em> caused an increase in the exports of the non-SLA<\/em> provinces rather than decrease in exports of SLA <\/em>provinces.<\/p>\n\n<\/div><\/div>\n
Example 2: Card and Krueger (1994)<\/a><\/h3><\/p>\nOn April 1, 1992, New Jersey raised the state minimum wage from $4.25 to $5.05. However, right across the Delaware river, in Pennsylvania, minimum wages were kept unchanged at $4.25. Thus, if a researcher wanted to study the causal impact of minimum wage on employment, then comparing the difference in employment between New Jersey and Pennsylvania before and after April 1992 would be ideal for a DID. Card and Krueger (1994)<\/a> did precisely that by studying the impact of the statutory minimum wage increase on employment at fast food restaurants where most minimum wage workers are employed. In practice, this is implemented in a simple panel regression with state and time fixed effects along with an interaction term:<\/p>\nYist<\/sub> = \u03b1 + \u00b5Dt<\/sub> + \u03c6ds<\/sub> + \u03b2Dt<\/sub>ds<\/sub> + uist<\/sub> ,<\/p>\nwhere Yist<\/sub> is the employment at restaurant i <\/em>in state s <\/em>at time t<\/em>; Dt<\/sub> is a time-dummy variable, which takes the value 1 for periods after the minimum wage increase and takes the value 0 otherwise; and ds<\/sub> is a state-dummy variable, which takes the value 1 for the treated state (New Jersey) and 0 for the control state (Pennsylvania). The key coefficient of interest is \u03b2, the coefficient of the interaction between the state and time dummies, as it is interpreted as the causal effect of a minimum wage increase on employment. To understand why \u03b2 is the so-called \u201cDifference in Difference\u201d estimate, a quick look at the following table of the conditional expectation of Y<\/span>ist<\/sub> will suffice:<\/p>\n
On April 1, 1996, Canada and U.S. entered into a five-year contract called the Softwood Lumber Agreement (SLA<\/em> henceforth), whereby a tariff rate quota was imposed on US-bound lumber exports from four Canadian provinces: Alberta, British Columbia, Ontario and Quebec. The first 14.7 billion board feet of softwood lumber from these provinces was exported duty free but further exports were subject to tariffs. A tariff of $50 per thousand board feet was imposed on the next 650 million board feet exported and further exports were going to be taxed at $100 per thousand board feet. The SLA <\/em>was novel since exports from only four provinces were limited and other provinces were not subject to any restriction.<\/p>\n Gulati and Malhotra<\/a> (2006) investigated firstly, whether the SLA<\/em> caused a reduction in softwood exports to the U.S. from the four provinces and if so, what was the size of the decline. Secondly, what was the size of the increases, if any, in the softwood lumber exports of the other provinces to the U.S.<\/p>\n The regression equation they run is the following:<\/p>\n Xit<\/sub> = \u03b10<\/sub> + \u03b11<\/sub>*Yit<\/sub> + \u03b12<\/sub>*YUS,t<\/sub> + \u03b13<\/sub>*Distt<\/sub> + \u03b14<\/sub>*Ext<\/sub> + \u03b15<\/sub>*RUS,t<\/sub> + \u03b16<\/sub>*SLA<\/em>i<\/sub> + \u03b17<\/sub>*Restt<\/sub> + \u03b18<\/sub>*SLA<\/em>i<\/sub>*Restt<\/sub> + ut<\/sub><\/p>\n and the STATA<\/abbr> command will be:<\/p>\n reg<\/strong> X Yi<\/sub> YUS<\/sub> Dist Ex R SLA<\/em> Rest SLA#<\/strong><\/em>Rest<\/p>\n where, Xit<\/sub> is log value of exports or log quantity of exports (annual) from province I to US. Yit<\/sub> and YUS,t<\/sub> are the log GDP of province i and US at time t respectively to control for demand of lumber in Canada and US. Disti<\/sub> is the log of distance from province I to the US border. RUS,t<\/sub> is the US interest rate to control for influence of interest rates on demand for new homes (a major source of softwood lumber). Ext<\/sub> is the US-Canada interest rate to control for price effects since exchange rates affect the relative price of Canadian lumber. SLA<\/em> is a dummy variable which takes the value 1 for provinces on which tariffs were applied under SLA<\/em> and 0 otherwise. Rest is a dummy for the years SLA<\/em> was in effect and 0 otherwise. SLA<\/em>*Rest is an interaction term that takes the value 1 for provinces under SLA<\/em> for the years SLA<\/em> was in place.<\/p>\n The key coefficient of interest is \u03b18<\/sub> as it is interpreted as the causal effect of the SLA on the change in exports of the SLA<\/em> provinces compared to the non-SLA<\/em> provinces. To understand why \u03b18<\/sub> is referred as the \u201cDifference in Difference\u201d estimate, take a look at the following table which is Table 4 in the paper.<\/p>\n \u03b16<\/sub> captures the difference in the mean of log exports for SLA<\/em> and non-SLA<\/em> provinces before the SLA <\/em>restrictions came into effect. \u03b17<\/sub> captures the effect of the SLA <\/em>agreement on the non-SLA<\/em> provinces. \u03b17<\/sub> + \u03b18<\/sub> reflects the effect of the SLA<\/em> contract on the provinces named in the contract. Thus, \u03b18<\/sub> shows the difference in export performance of the provinces named in the SLA <\/em>vs. not included in the SLA<\/em> because of the SLA<\/em> restriction.<\/p>\n Their results indicate that the SLA<\/em> had a significant impact on the exports of non-SLA<\/em> provinces and the SLA<\/em> by itself increased the exports from these provinces compared to the SLA<\/em> provinces by four times. But, the SLA<\/em> lead to a decrease in exports of SLA<\/em> provinces of only 5 percent which was statistically insignificant. Thus, SLA<\/em> caused an increase in the exports of the non-SLA<\/em> provinces rather than decrease in exports of SLA <\/em>provinces.<\/p>\n \n<\/div><\/div>\n On April 1, 1992, New Jersey raised the state minimum wage from $4.25 to $5.05. However, right across the Delaware river, in Pennsylvania, minimum wages were kept unchanged at $4.25. Thus, if a researcher wanted to study the causal impact of minimum wage on employment, then comparing the difference in employment between New Jersey and Pennsylvania before and after April 1992 would be ideal for a DID. Card and Krueger (1994)<\/a> did precisely that by studying the impact of the statutory minimum wage increase on employment at fast food restaurants where most minimum wage workers are employed. In practice, this is implemented in a simple panel regression with state and time fixed effects along with an interaction term:<\/p>\n Yist<\/sub> = \u03b1 + \u00b5Dt<\/sub> + \u03c6ds<\/sub> + \u03b2Dt<\/sub>ds<\/sub> + uist<\/sub> ,<\/p>\n where Yist<\/sub> is the employment at restaurant i <\/em>in state s <\/em>at time t<\/em>; Dt<\/sub> is a time-dummy variable, which takes the value 1 for periods after the minimum wage increase and takes the value 0 otherwise; and ds<\/sub> is a state-dummy variable, which takes the value 1 for the treated state (New Jersey) and 0 for the control state (Pennsylvania). The key coefficient of interest is \u03b2, the coefficient of the interaction between the state and time dummies, as it is interpreted as the causal effect of a minimum wage increase on employment. To understand why \u03b2 is the so-called \u201cDifference in Difference\u201d estimate, a quick look at the following table of the conditional expectation of Y<\/span>ist<\/sub> will suffice:<\/p>\n<\/p>\nExample 2: Card and Krueger (1994)<\/a><\/h3>