What is the proper amount of hay per day to feed a cow during the winter?

This is a loaded question because it depends on a number of factors, namely:

1. Cow body weight
2. Body condition score (is she thin, fat, or has moderate fat covering)
3. Reproductive status (lactating? pregnant & what stage of pregnancy?)
4. Breed type (Angus or British cross, Simmental/dairy-type?)
5. Temperature and wind
6. Coat condition (dry clean, muddy, or snow covered)
7. Hay quality (protein and TDN levels?)

So, to get things rolling the best thing to do is provide an example cow to give an example winter ration using hay.

Let's say this is a 1400 lb dry, bred cow in mid-pregnancy. It's January 15th and current air temperatures are sitting at -20ºC; last month was the same, and she has a winter coat on that's clean and dry. She has a moderate body condition score of 5 (out of a scale of 1 to 9, with a BCS of 1 being truly emaciated), so she's in good condition. She's sitting where she needs at least 7 to 8% protein. She's an Angus-Simmental cross, with most of her breeding as Angus. (What bull she's bred to has no influence on her nutritional requirements.) The hay is good quality grass hay, tested at around 9% protein and with a TDN value of 60%. Dry matter content is 18%.

Since she's at 1400 lbs, she's expected to eat, at the minimum, 30 pounds of hay per day. At the most, she'll consume around 35 to 38 pounds per day, as dry matter ration. So, according to my calculations, the proper amount to feed her per day is around 35 pounds per day. She'll be getting enough protein from the hay alone so there's no need to worry about feeding extra grain until she calves and is lactating. A lactating beef cow will need 11% protein or more, so bringing the hay down by only two or three pounds and adding ~5 pounds of barley or oats will be enough to give her enough protein and energy for milk production.

The reason I added those seven factors was that each of these factors will influence how much a cow will eat per day. Here's how:

1. Cow body weight: A big cow will eat more than a small cow. Both will have the same requirements in terms of body weight at 2 to 3% body weight in dry matter ration per day, but the amounts will change. For example, a 1000 pound cow eating 2.5% of her body weight per day will consume 25 pounds of DM ration daily minimum, whereas a 1400 pound cow will consume 35 pounds of DM ration daily minimum. (DM = dry matter = all water removed) The same rests with comparing a mature cow to a growing yearling calf: The cow will eat more.
2. Body condition score (is she thin, fat, or has moderate fat covering): A thin cow will need to eat more than a cow in moderate condition or that is fat. This is because her metabolism is increased to compensate for changes in temperature. A thin cow is more sensitive to temperature than a fat cow is, so if it gets colder, she needs to eat more to gather enough energy to keep warm and to maintain herself. Better quality feed needs to be given to thin cows also so they gain weight.
3. Reproductive status (lactating? pregnant & what stage of pregnancy?): A lactating cow needs more feed (and better quality) than a dry, pregnant cow because she's feeding both herself and her suckling calf. Her nutritional needs are also increased; where a dry cow in mid-pregnancy only needs a minimum of 7% protein, a lactating cow requires 11% protein or more. Energy needs also increases in a similar manner. Nutrition needs peak for lactating cows when they reach about the 3rd month of lactation post-calving.
4. Breed type (Angus or British cross, Simmental/dairy-type?): Typically British and Continental breeds, except Simmental, are less demanding for nutrition than Simmentals and dairy-type cattle. Dairy-type cattle like Holsteins and Jerseys need more higher-protein and higher-energy feeds to meet their increased metabolisms. We can see an increase in requirements between the two groups by as much as 10%.
5. Environmental temperature and wind: The colder the temperature, the more a cow will eat to produce enough heat via their rumens to keep warm. Cold stress is a threat in winter time, so cows will need more energy and more feed when temperatures dip below their lower threshold limit of -20ºC (assuming they're in good condition). Wind chill can really cause problems, and also play a role in feed consumption; the more the wind blows, the more feed they'll eat and more energy they'll need to keep warm. A good rule of thumb to remember is that for every drop in 10 degrees below -20ºC, add 2 pounds of grain to their current ration. A lactating cow in cold weather will need 40 to 60% more energy in cold weather than a dry, pregnant cow.
6. Coat condition (dry clean, muddy, or snow covered): In the case of winter time, a cow with snow-covered and wet coat or matted and muddy coat will eat more and need more in terms of energy and somewhat protein than a cow with a dry-clean coat. A cow out in the cold that still has her summer coat on will also be needing to eat more and have increased energy requirements. Winter coats provide some protection from the elements, so long as they're dry and clean. Highland and Galloway cattle have a bigger advantage over Angus or Hereford cattle because they have heavier, denser coats, thus require less feed and less energy. Skin thickness has no effect on nutrient requirements in cold weather.
7. Hay quality (protein and TDN levels): Generally, the poorer the hay, the less of it a cow will eat, without an added protein supplement. This is because the lowered protein content slows the rate of digestion which will make a cow feel full more often. The microbes in the rumen have a tougher time breaking down the feed, meaning they take longer to break down the feed with less protein coming from that feed. The opposite is true with better quality hay. This is why protein and/or grain supplementation is important for times when cows need them most.