* Unemployment edges down to 26.3 percent in second quarter
* Three banks, Telefonica, Repsol give encouraging trading updates
* Data strengthens government's view economy can grow in Q3
By Paul Day and Sarah White
MADRID, July 25 (Reuters) - Spain's unemployment rate fell for the first time in two years and some of the country's biggest firms said on Thursday business was looking up, boosting the government's claim the economy is climbing out of recession.
The dip in the jobless figures - to 26.3 percent in the second quarter from 27.2 percent in the first - nonetheless highlighted how far the country still needs to travel on the road to full recovery. Economy Minister Luis de Guindos called the size of the figures "totally unacceptable".
Spain is one of the euro zone's troubled economies, subject to a bank bail out after a housing bubble burst. But it is far stronger than others such as neighbour Portugal or Greece.
Jobs have been the main issue. The unemployment rate has risen relentlessly since 2011, with some 3.8 million people joining the jobless lines since the first quarter of 2008, the year the global financial crisis erupted and property prices collapsed.
The real estate meltdown left the country's banks heavily exposed to soured assets and loans, which have since weighed on their balance sheets and soaked up 42 billion euros ($55.59 billion) of European Union aid.
On Thursday, three banks including bailed out lender Bankia , reported bad debts were still rising. But they also posted big jumps in first-half profits, on lower writedowns on property assets and trading gains.
Telecoms firms Telefonica and oil major Repsol - two heavyweight Spanish firms with more exposure to foreign economies - also gave encouraging trading updates.
Supported by a central bank report earlier this week showing Spain's economy came close to stablising in the second quarter, Thursday's run of encouraging news added weight to Prime Minister Mariano Rajoy's belief the economy should exit its two-year recession as soon as the current quarter.
"Even seasonally adjusted (unemployment) data is better than we expected, which is in line with the economic improvements forecast by the Bank of Spain," Angel Laborda, economist at think tank Funcas, said.
De Guindos repeated the forecast for third quarter growth on Thursday and said he was convinced the worst was over for the economy, but Rajoy had no chance to share in his economy minister's cautious optimism.
The prime minister focused on a visit to his home province of Galicia to meet survivors of a train derailment late on Wednesday that killed least 78 people on the eve of the one of the country's biggest religious festivals.
Madrid passed a series of austerity measures last year to tame one of the euro zone's highest public deficits.
That hobbled already depressed domestic demand, but has also taken it further than some of its peers along the path of economic reforms mandated by Brussels to get the euro zone economy back on its feet.
Spain has also won tacit agreement this year from its euro zone partners to ease off on tax hikes and spending cuts.
"The Spanish government has undertaken a bitter battle to stabilise growing unemployment, and it seems to be gaining ground," Nancy Curtin, chief investment officer of Close Brothers Asset Management, said in a written note.
"But despite positive data in recent weeks, it's evident that Spain is still struggling to balance austerity with growth."
Despite government projections, many economists believe the country's second recession in three years is unlikely to end in 2013, and joblessness remains the weakest link.
Seasonal tourism accounted for most of the drop in the latest unemployment rate, and the sector is expected to be strong this year as cash-strapped Europeans look for budget vacation spots while avoiding Egypt and other Middle Eastern troublespots.
But Spain, along with Greece, still has a far greater share of its population out of work than other euro zone states.
Around half of Spain's near six million unemployed have not held a job for more than one year, while 1.8 million homes have no one in work, the data showed.
After a decade of above average economic growth, the long recession prompted hundreds of thousands to leave the country in 2012, including immigrants returning home and Spaniards in search of work elsewhere.
Telecoms leader Telefonica, which employs more than three-quarters of its 132,000-strong workforce abroad, said on Thursday it cut borrowings to under 50 billion euros ($66 billion) in the first half.
That put it on track to meet full-year targets even after the planned purchase of German mobile operator E-Plus, though earnings and revenues fell due to pressures in its home market.
Profits at oil major Repsol rose more than expected as strong production offset weak refining margins. New upstream projects have helped the company recover steadily from the nationalisation of its majority stake in Argentine energy company YPF in May of 2012.