This question studies whether working 4 days a week (therefore not working on Fridays) would increase or decrease the productivity of workers, using a hypothetical scenario that New South Wales has implemented a law that mandates working 4 days a week in 2030. You collected data on the productivity of workers in three waves, in 2025, 2030, and 2035:

State	Year	Work4	Hourly Rate	Average Years of Education
New South Wales	2025	0	40	13
New South Wales	2030	1	42	13
New South Wales	2035	1	46	13
Victoria	2025	0	41	15
Victoria	2030	0	43	15
Victoria	2035	0	45	15
Queensland	2025	0	30	12
Queensland	2030	0	35	12
Queensland	2035	0	40	12

Description for this dataset are as follows:

• State: select states in Australia
• Year: the year that data is collected
• Work4: =1 if the law mandates working 4 days a week, =0 if the law mandates working 5 days a week
• Average Years of Education: average years of education of workers in the state

(a) Using hourly rate and average years of education data in 2025, find an optimal rule, by minimizing the sum of squared errors, for synthesizing New South Wales using Victoria and Queensland.

(b) Using the optimal synthesizing rule in (a), calculate the causal effect of working 4 days on productivity (i.e. hourly rates) of New South Wales workers in 2030.

(c) Using the optimal synthesizing rule in (a), calculate the causal effect of working 4 days on productivity (i.e. hourly rates) of New South Wales workers in 2035.

(d) Compare causal effects in (b) and (c). Which is larger? What does it tell us about the short-term and long-term effects of working 4 days a week?